Ilkay Gundogan has signed for Barcelona on a free transfer at the end of his Manchester City contract while Luka Modric will stay at Real Madrid for another season after renewing his deal until June 2024.

The German midfielder Gundogan, 32, has agreed a two-year deal with the option for a year's extension and a release clause of 400 million euros ($436 million).

"He is bringing his presence, nose for goal and reading of the game which has made him one of the best midfielders of the past decade," said Barcelona in a statement.

Gundogan has been an influential presence for Pep Guardiola's City side since joining in 2016, captaining the team to a stunning treble this season.

The midfielder netted twice in the FA Cup final win over rivals Manchester United and Guardiola admitted City were trying to renew his deal.

Gundogan joined City from Borussia Dortmund, where he won the Bundesliga and German Cup in 2012.

In Manchester he scored 60 goals in 304 appearances for City.

Meanwhile, Modric, 37, decided to stay at Real Madrid despite strong interest from Saudi Arabia, looking to recruit him at the end of his previous contract, which expires this month.

The Croatian follows Toni Kroos, Dani Ceballos and Nacho Fernandez in extending his time at Madrid this summer.

"In his 11 seasons defending our shirt, he has played 488 games and won 23 trophies," said Real Madrid in a statement.

Modric has lifted the Champions League five times with Madrid since joining in 2012 from Tottenham, as well as La Liga on three occasions and two Copas del Rey.

Turning 38 in September, 2018 Ballon d'Or winner Modric faces stiff competition next season at the Santiago Bernabeu.

Madrid signed England international Jude Bellingham from Borussia Dortmund for over 100 million euros ($109 million) earlier in June, while Kroos, Ceballos, Eduardo Camavinga, Aurelien Tchouameni and Fede Valverde are jostling for places.

Youngster Bellingham said at his presentation he was looking forward to learning from Modric and Kroos in particular.