Alcantara is believed to be considering his future, and has an offer on the table from Manchester United who want to wrap up the deal before their preseason tour of Australasia gets underway in two weeks' time.
It was also claimed yesterday that the Spain U21 star's father returned from Manchester yesterday and will tell Barcelona that his son has his heart set on a move to Old Trafford.
However according to the Times, Barcelona remain hopeful of convincing him to stay put with the promise of more regular first team football next season.
Alcantara, who scored a hat-trick for Spain in their 4-2 win against Italy in the U21 European Championship final in Israel, has a release clause of around €18 million in his contract because of a lack of game-time at the Camp Nou.
The Times add that the young playmaker has agreed terms on a five-year deal worth around £120,000-per-week - quadruple what he earns at Barcelona - and coupled with the promise of first-team action, the Red Devils believe that they have done enough to get their man.
In their attempts to ward off interest from Manchester United, Barcelona are ready to offer Alcantara, who wants first-team football to secure a spot at the World Cup next year, a long-term lucrative deal and the promise that he is the long-term successor to Xavi.
Should Manchester United be succesful in thier attempt to land Alcantara, the Times claim a move to bring Cristiano Ronaldo back to Old Trafford could still be on the cards.