United confirmed last Monday that they had made a first offer for the former Arsenal star, and manager David Moyes said at a press conference in Yokohama on Monday that a second offer had been lodged.
Executive vice-chairman Ed Woodward returned to the UK ahead of schedule last Wednesday to carry out some urgent transfer business, and Moyes confirmed: "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 Uruguay Under-20 star Guillermo Varela.
In addition to Fabregas and Cabaye, 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.