The Barcelona midfielder has repeatedly said he wants to stay at his hometown club despite difficulty winning over some supporters last season.
That was his second year back in Catalonia following his return from Arsenal, who reportedly have first refusal on Fabregas if Barca decide to sell.
United's offer is believed to be below Barcelona's valuation of the Spain international.
United are searching for a creative midfield presence to offset the loss of newly-retired Paul Scholes.
Fabregas would become the second former Arsenal captain at Old Trafford, alongside Robin van Persie, if the Premier League champions managed to pull off a deal.
Speaking last month, Fabregas confirmed he is happy at Barcelona and dismissed talk of his departure by reiterating his desire to stay at the Camp Nou.
"I've always said that I'm happy at Barca. Whoever says that I'm leaving doesn't know me and has never spoken with me," said the 26-year-old.
"Nor with my agent, because he knows that I don't want him to talk with other people about me. I'm very private and I don't even talk about these things with my family.
"If someone says anything that isn't that I want to play at Barca, it's completely absurd."
New United manager David Moyes is still waiting to make his first signing after missing out on Thiago Alcantara after the midfielder left Barcelona for a reunion with Pep Guardiola at Bayern Munich.
However, Moyes says he has money to spend this summer and that the Red Devils are looking to land 'the best players' around.
After seeing his first game at the United helm end in a 1-0 defeat to the Singha All Star XI on Saturday, Moyes said: "Isn't it great that the club says, 'There's no budget here, you go get who you want to get, just go and do it'.
"We are looking at the best players. I do think it's important that we show people we are carrying on the traditions and trying to take the club forward.
"I will not name anybody, but certainly we are always looking at the best players. I think we are trying to strengthen in a couple of areas. We are short in midfield.
"We are also a club which tends to invest in the youth side, so you can expect us to go for players in that age group.
"We are working hard to make sure the first signing is the right one. I have been left a really good team and I want to add to it sensibly."