Arsenal have confirmed defender David Luiz has left the club while Martin Odegaard and Dani Ceballos have returned to Real Madrid.

Experienced defender David Luiz indicated he would be leaving the Gunners at the end of his contract when he posted a farewell message to fans on Instagram last month.

The 34-year-old, who joined from Chelsea for a reported £8million in August 2019, made 73 appearances in all competitions for Mikel Arteta's side.

Meanwhile, Odegaard and Ceballos are heading back to Madrid following the expiry of their loan spells at Emirates Stadium.

Odegaard, who joined in January, impressed in his 20 appearances in all competitions under Arteta.

The Norway international scored two goals and provided two assists, while his average of 2.24 chances created per 90 minutes was the best return of any Arsenal player to play more than six times last season.

Ceballos first signed for Arsenal on loan in July 2019 and the deal was extended last September after an impressive campaign that began with two assists on his first Premier League start, a feat only previously achieved for Arsenal by Ray Parlour and Henrikh Mkhitaryan.

The 24-year-old midfielder made 40 appearances for the Gunners last season, although only 25 of those came as a starter.

Each player has been linked with a permanent move to the club, but new Madrid coach Carlo Ancelotti name-checked Odegaard and Ceballos this week as he discussed their trophy hopes in 2021-22.

"With this squad, without loan players like [Gareth] Bale, Odegaard and Ceballos, Madrid reached the Champions League semis and fought for LaLiga. So, we have the players to win trophies," Ancelotti told ABC.

Goalkeeper Mat Ryan is also leaving Arsenal after a short-term loan, while youngsters Levi Laing, Joseph Olowu, Luke Plange and Jason Sraha have been released.

In a statement, Arsenal said: "All developments regarding players extending their contracts; joining-rejoining us; or leaving us will be officially communicated as soon as they are confirmed.

"We thank the above players for their contributions to the club during the periods of their permanent or loan contracts with us.

"You will always be part of the Arsenal family and we wish you all the best of health, happiness and success in your futures."

Dani Ceballos has made clear he does not want to be loaned out again by Real Madrid, despite learning plenty during his two seasons with Arsenal.

Midfielder Ceballos won the FA Cup at the end of the 2019-20 campaign under Mikel Arteta, his first on loan at the Gunners. He returned to Emirates Stadium again, agreeing to another season-long stint in England.

However, having played 77 games for Arsenal, the Spain international now wants his long-term future resolved one way or another, revealing he wants to "feel important" at a club.

He is under contract for two more years at the Santiago Bernabeu, having joined Madrid from LaLiga rivals Real Betis in July 2017.

"It is true that I have a contract with Real Madrid for the next two years and I would not like to go on loan to any club again,” Ceballos said on the El Pelotazo show on Canal Sur Radio, according to Marca.

"With the experience of the last two years, I am a player who needs to feel important, I would like next year starting from the beginning and feeling important, but settling down.

"It's important going forward to know what you want to do."

Zinedine Zidane's departure could provide a fresh start for Ceballos at Madrid, who are now looking for a new head coach to replace the Frenchman.

On the change at Madrid, Ceballos said: "It is true that with Zizou I had fewer opportunities than I had thought, but now Madrid is without a coach, we do not know what will happen in two or three weeks, there is talk of many things and I think that right now is time to disconnect a bit from football, prepare well for what is coming and when Madrid have their next coach, let's talk and see what we want, it will be time to decide the future.”

Ceballos started 25 games for Arsenal in the 2020-21 season, with his final total of minutes (2,370) a fraction above the previous term (2,343).

The 24-year-old provided three assists and also created six big chances – deemed by Opta as those situations where a player should score – though he did not get a goal himself, having managed two in 2019-20.

"The Premier is a very attractive league, I was lucky to be able to play it last year and win the FA Cup, and this second year has helped me to consolidate myself," Ceballos said of his time at Arsenal.

"I am very happy because I have learned a lot, I have become a better player, more complete, and I think that although my stage has not closed as we would have liked, with another title, I am very happy with the experience."

Dani Ceballos knows only too well why Martin Odegaard has become frustrated at Real Madrid, insisting ambitious young players "need to play regularly" to further their development.

Odegaard impressed last season during a loan spell at LaLiga rivals Real Sociedad, yet he has struggled for minutes this term having returned to Madrid.

The Norway international has started just five games in all competitions and clocked up a mere 233 league minutes, leading to reports in Spain that he wants to leave in the January transfer window.

Ceballos himself is spending a second season on loan at Arsenal, a decision taken to make sure he had a greater opportunity for first-team action.

"I don't know exactly what happens with Martin, but it is obvious that any player who wants to succeed at Madrid is going to grab the opportunity he has," Ceballos said in an interview with Efe.

"I don't know what Martin thinks, but I know I think that when you're young and you have goals and objectives, what you need is to play regularly.

"If you don't, it's like something is missing. I guess that's what's happening to Martin."

Odegaard has failed to score or assist in his nine appearances for Madrid this season, having contributed seven goals and nine assists in 2019-20 while away on loan.

The 22-year-old has been linked with a return to Sociedad, potentially in a loan deal that includes an option to buy.

Ceballos, meanwhile, has not given up hope of forcing his way in at Madrid, though will need assurances over his future role if he is to remain in the Spanish capital next season.

"They don't have to tell me anything," the midfielder - who has won 11 caps for Spain - said of his own future. 

"It would be to have a talk with the coach and the coaching staff that is in at that moment and know that I will have equal opportunities with him.

"From there, you have to earn it on the field."

Ceballos has played in 24 games for Arsenal this term under Mikel Arteta, his most recent start coming in the 0-0 home draw with Crystal Palace on January 14.

Dani Ceballos says his priority is to become a regular for Real Madrid once his second season on loan at Arsenal has come to an end.

The midfielder impressed for the Gunners in 2019-20 before extending his loan stay for an additional season.

He has not played as regularly in this campaign, prompting speculation Madrid would recall the 24-year-old and loan him out elsewhere to get more minutes.

But Ceballos did return to the Arsenal starting XI with an impressive performance in the 4-0 away win over West Brom last time out, the club's third consecutive triumph.

While Ceballos is loving playing under Gunners boss Mikel Arteta, he still sees his future with Madrid rather than in London.

"A third year on loan? No, my goal is to return to Real Madrid and be important at the best club in the world," he told El Larguero.

"I believe that one day I will come back and be able to return the affection that the fans showed me.

"I signed a long contract with Madrid. I have two-and-a-half years left. 

"For various reasons, I didn't leave the club in the best manner. My goal is to succeed at Arsenal [but] my wish is to wear the Madrid shirt.

"For those of us from Madrid, the arrival of Zinedine Zidane has been good for everyone. 

"Each coach has their taste in players. He told me that he needed me to play and get minutes. He was clear with me."

Discussing life under Arteta with Arsenal, Ceballos, who has started nine of the Gunners' 17 Premier League matches, continued: "Mikel is a coach who suits me like a glove.

"That is because he played in the same position as me. He wants us to play forward when we steal the ball. 

"The Premier League is a bit below [LaLiga] at a tactical level but it is more physical. The position of number eight is the one that suits me best.

"We started to get into bad dynamics and when a team has a priority to fight for Europe it is difficult to get out. We have chained three victories together and hopefully we will go up [the table]."

Ceballos also said he would love to be reunited with the Real Betis fans by playing there again later in his career.

"Betis is different from any club in the world," he added.