The Catalan giants have said publicly that they do not want to sell the former Arsenal star.
However a source close to Fabregas has told Sky Sports News that the player is considering his position having been given no assurances that he will be a regular starter next season.
Barcelona may consider a £35million offer, according to the source, but will ultimately leave the decision up to the player and will not force him out.
A final decision on Fabregas' future could now be accelerated, though, after Barcelona appointed Argentinian Gerardo Martino as their new coach in place of Tito Vilanova.
Manager David Moyes stated on Monday that he was waiting for a response and that chief executive Ed Woodward was dealing with the bid.
Speaking at a press conference in Yokohama, Moyes revealed: "As I understand it, Ed had a response (to the first offer). We have made a second offer but Ed is dealing with it rather than me."
So far, United have not made any new signings this summer, other than Uruguay Under-20 star Guillermo Varela and, whilst Moyes accepts there may come a time when he has to walk away from the deal, that position has not been reached yet.
"I think a point does come," said Moyes. "But when you are interested in good players you want to give it every opportunity to materialise.
In addition to Fabregas, Moyes has also been linked with marquee signings Cristiano Ronaldo and Gareth Bale, plus Everton duo Leighton Baines and Marouane Fellaini, so far though, there have been no reinforcements and Moyes admitted he can understand why some fans are nervous.
"I can understand that feeling," he said. "But there have been quite a few changes. Other managers have changed their clubs too and everybody thought this might be the summer when transfers are done later in the month.
"That was always the thought I had when I came into the club. You have to remember I only started on July 1, so in truth I have only been in the job three weeks at this present time."
"I will hope things will continue to move forward. At this moment in time I can only tell you Ed Woodward is working hard to try and make the deals happen. We hope some of them will fall into place shortly."