United confirmed last Monday that they had made an initial approach for the former Arsenal star, but that offer of £25 million ($38 million, 29 million euros) was swiftly rejected by the Spanish champions.
Moyes is determined to do everything he can to lure Fabregas back to the Premier League as he seeks a marquee signing to mark his arrival at Old Trafford.
And the former Everton boss, appointed as Alex Ferguson’s successor at the end of last season, said at a press conference in Yokohama on Monday that a second improved offer, believed to be in the region of £30 million ($46 million, 34 million euros), has now been lodged.
United executive vice-chairman Ed Woodward returned to England ahead of schedule last Wednesday to lead the Fabregas negotiations and Moyes said: “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.”
Moyes has yet to bring in a major signing, with his only deal so far the capture of Uruguay Under-20 star Guillermo Varela.
But, with reports from Spain suggesting Barca will not make a decision on Fabregas until they appoint a successor to coach Tito Vilanova, Moyes acknowledged there may come a time when he has to turn to other targets.
“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.”
Moyes has also been linked with moves for Real Madrid forward Cristiano Ronaldo and Tottenham winger Gareth Bale, plus raiding his old club Everton for Leighton Baines and Marouane Fellaini.
So far though he has been frustrated, with new Madrid boss Carlo Ancelotti on Sunday making it clear he doesn’t expect Ronaldo to rejoin United.
And Moyes, who is also fighting to persuade Wayne Rooney to snub advances from Chelsea, admitted he can understand why some fans are nervous about the lack of fresh faces as the champions gear up for the first season of the post-Ferguson era.
“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.”