Moyes told a news conference in Yokohama today that a second offer has been made for Fabregas.
The new bid is believed to be around £30million for the former Arsenal star.
“As I understand it, Ed had a response [to the first bid],” said Moyes at a press conference in Yokohama.
“We have made a second offer but Ed is dealing with it rather than me.”
The Scot added: “When you’re interested in good players you want to give it every opportunity to materialise and I’ll do that. I hope things can continue and move forward.
“At this moment in time I can only tell you that Ed Woodward is working hard trying to make the deals happen. We can only hope that some of them fall into place shortly.”
United had a first offer of around £26million rejected by Barcelona last week.
The pressure is on new manager Daivd Moyes to make his first meaningful signing this summer.
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Liverpool have unveiled their new home kit for next season, sticking with the hugely popular retro styling of the current shirt.
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