United had an initial £25million bid turned down by the Catalan club last week but Moyes has submitted a new £30million offer that is set to test the resolve of both Barcelona and Fabregas.
Tito Vilanova stepped down as Barca boss on Friday after battling with cancer for the past few months.
The Spanish champions want to keep the former Arsenal midfielder but after becoming more of a substitute last season, Fabregas may well want a return to the Premier League where he will undoubtedly be a key player for United.
Arsenal have confirmed they will not be bidding for their former player leaving United as the sole interested party.
Barcelona may wait until a new manager is in charge before accepting or rejecting any bids but United will move on to other targets if they remain unsuccessful with Fabregas.
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