Manchester United reportedly registered a formal bid in the region of £25m for the Spanish international yesterday, with the Catalan giants yet to comment on speculation surrounding the midfielder's future.
Keown thinks that Fabregas should resist the temptation to move back to England with either United or former club Arsenal, because he has unfinished business with Barca.
He told the Daily Mail: "Fabregas went there to take over from Xavi and Andres Iniesta and that can still happen. He's had injuries but he's still an outstanding talent. I saw that during the Confederations Cup this summer.
"Yes, a move to Manchester United or back to Arsenal might feel tempting but I think he should stay in Barcelona. It was his dream for such a long time. It's character building to carry the fight through.
"He'd always feel a tinge of regret if he left, even though he's won the league and cup already."
United boss David Moyes has done little to dampen speculation that he wants to sign Fabregas this summer, publicly declaring that he wants to strengthen in midfield.