United confirmed last Monday that they had made a first offer for the former Arsenal star, and manager David Moyes said at a news conference in Yokohama on Monday that a second offer had been lodged.
Ed Woodward, the executive vice-chairman, returned to the UK ahead of schedule last Wednesday to carry out some urgent transfer business, and Moyes confirmed on Monday: "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 Guillermo Varela, the Uruguay Under 20 star.
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.
"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."
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 has been no reinforcements.
And Moyes conceded 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."
Moyes confirmed he expects Michael Carrick, Jonny Evans and Alex Buttner will have recovered from minor injuries in time to face J League outfit Yokohama F Marinos on Tuesday.
And Chris Smalling and Ashley Young could also play some part after recovering from foot and ankle injuries respectively, while Japan star Shinji Kagawa has also joined up.
"They probably won't start in this game but I want to get them involved," said Moyes of the latter trio.
"Young and Smalling have been training very hard back home and Shinji has been on holiday but he has been training.
"We will involve the three players as we go along."
For Kagawa it is the start of an important season, having failed to make the impact he hoped after a £17 million (Dh95.7m) switch from Borussia Dortmund 12 months ago.
That Dortmund subsequently went on to face Bayern Munich in the Champions League final could have left Kagawa believing he had made a mistake coming to Old Trafford, but he insists that is not the case.
"I was proud to see my former team play in the Champions League final, and I was very excited for them," he said (see gallery).
"But it didn't make me feel I had made the wrong choice.
"In fact it made me feel stronger because I want to be on the same field in future."
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