Moyes has identified the need for two midfielders, at least one of whom must be able to contribute goals, which Fábregas has proved he can do in the Premier League when at his former club, Arsenal. Fellaini also has an eye for goal; he scored 12 times for Everton last season.
Moyes considered Real Madrid's Luka Modric before deciding on lodging an official offer for the former Arsenal captain.
The deal hinges on whether Barcelona – who purchased the player from Arsenal for €29m (£25m) in 2011 – are willing to sell. Arsenal have a first-option clause on Fábregas should Barcelona make him available but this depends on him wanting to return there, which is unlikely.
Publicly Barcelona have maintained the midfielder is not for sale – "the player is not transferable," a source said – but any indication that they would consider allowing him to leave would force Fábregas's hand as it would be clear that he is not considered central to the plans of Tito Vilanova, the Barcelona coach. However the news of United's bid came as a surprise to Fábregas, so the player's thoughts are not yet known.
Moyes has turned his attention to the 26-year-old after Thiago Alcântara joined Bayern Munich from Barcelona for £21.6m. Yet it is thought that Moyes could have signed Thiago over the course of the past weeks, with the delay in his finally leaving for Bayern being down to United's manager not being completely convinced by the 22-year-old. Although it is understood Thiago was keen on joining United, there were suggestions from some at the club that he might struggle on occasion to cope with the physicality of the Premier League.
Last month Fábregas said he wanted to stay at Camp Nou. "I've always said that I'm happy at Barça. Whoever says that I'm leaving doesn't know me and has never spoken with me," he said. "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 Barça, it's completely absurd."
As first reported in the Guardian two months ago Moyes has long retained an interest in Fellaini, the Belgian he took to Goodison Park in 2008 for £15m from Standard Liège. Although no official bid has yet been made, the 25-year-old falls into the category of being is a genuine option for Moyes, who is now considering returning to his old club with an offer. Fellaini, whose contract runs until 2016, has a release clause of around £23m and a portion of any fee received by Everton would have to be paid to Liège. It is thought the release clause expires in mid-August.
Although Fellaini is far from the finished article, Moyes believes if he were to move to United he could become a high-class operator. The 6ft 4in Belgian would offer a physical presence currently lacking in United's midfield and, having operated in a deep-lying position and further advanced, is also versatile. As with Fábregas, Fellaini carries a goal threat from midfield, an area the Scot is keen to address.
Following Everton's first pre-season tour game, a 2-1 defeat by Austria Vienna in which Fellaini was captain for the second half, Roberto Martínez, the manager, talked up the prospect of him remaining at the club.
"He's still a very young man but I am really looking forward to seeing Marouane working hard and making sure that he sets really good standards," he said. "Marouane is an international footballer and he's going to be involved, if all goes well, in a World Cup next year."
Adnan Januzaj, United's 18-year-old attacking midfielder, is in talks with the club over a fresh contract, with the Belgian attracting interest from Barcelona and Juventus. Januzaj was United's stand-out player in the first half of Saturday's 1-0 defeat by the Singha All Star XI in the club's opening pre-season tour game in Bangkok, before he faded.
Moyes's predecessor, Sir Alex Ferguson, initiated the discussions with Januzaj, who was signed from Anderlecht two summers ago. After Saturday's defeat, Moyes said: "If he continues to play like that, he'll get more chances."
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