There are two separate stories among these. The first is that United have made a £25m bid. The second is that United will make a £35m bid. Of course, the two aren't mutually-exclusive, since the first, lower offer will almost certainly be rejected, but it's curious for a leaked story to go through the chinese whispers process.
A quick look at the sources - for AS, and possibly Sky too, it's likely to be Guillem Balague. Balague's record is extremely sketchy on these sort of things, but this report is backed up by pretty much everyone, and Sky don't go to their front page in the UK with 'Sky understands' based on Balague's hunches. We can pretty safely assume that this bid has either been made or is imminent.
Secondly, a note to cynics. Some have suggested that United are merely coping with the PR fallout over Thiago, assuring the club that we're after top players without much conviction of actually making the signing. This doesn't make a whole lot of sense - actually getting turned down by top players is far worse PR than not bidding for them in the first place, so it's difficult to see United making such a bold move if they didn't think they had a chance of succeeding.
I've written before about what a great fit Fabregas would be for United. After Thiago's departure, I think there are only four players out there even potentially available who could be a perfect fit in bringing exactly what we need - Cesc Fabregas, Luke Modric, Ilkay Gundogan, and Arturo Vidal. All of these would be transfers that the selling club are highly unlikely to want to sanction, regardless of whether the player actually wants to come. On the other hand, they're all good ages, and bring the perfect blend of control, guile and drive that our midfield so desperately lacks. That's why we should be happy to see United pay top money for them.
The suggested asking price for Cesc Fabregas by Barcelona is £34m. If United are prepared to deal at that price, I think it's a good one, and it's certainly going to be worth that to us. I don't think this is where our purchases should stop - we could do with a winger, a left-back, and certainly another midfielder in addition - but it would be perhaps the most important signing we've made in a very, very long time.