The Manchester Evening News say Premier League leaders City have told Ukrainian champions Donetsk that they will pay £45m to secure a January deal for Fred.
However, it is claimed City have been told that figure is below their valuation for a player who has also attracted attention from United and Arsenal.
City have moved for Brazil international Fred as Guardiola searches the market for a long-term successor to Fernandinho, who continues to impress but, at the age of 32, is reaching the twilight of his career.
Fred, 24, was believed to be identified by United as an ideal replacement for Michael Carrick, another player who is coming to the end of a fine career.
Reports from last week suggested United were joined by Arsenal at the front of the queue for Fred, but City have moved ahead of their Premier League rivals after contacting Donetsk.
Guardiola is also hoping to sign a centre back this month, with Jonny Evans a top target, but is ready to end his interest in Alexis Sanchez, appearing to give rivals United free run in the battle to sign the Chilean.
Speaking about United’s interest in Sanchez on Friday, Jose Mourinho said: ‘I don’t know if it is ethical or correct. The word we can find to be speaking about players of other clubs.
‘If somebody speaks of one of my players to tell we are interested or we are not interested, I wouldn’t be very, very happy.
‘Sanchez is an Arsenal player, I don’t know but probably this weekend he is going to defend Arsenal colours so I don’t think it’s correct to say things about Alexis Sanchez.
‘At the general level what I can say is myself and the Manchester United board, the owners, we don’t believe a lot in the January market, we don’t believe in signing a player just to sign a player, just to do something, we don’t believe in that.
‘What we believe really is there are some players in football world if you have the chance to sign them in January, in March or in July, you have to try and that’s it.
‘But in relation to Alexis Sanchez I don’t say a word, the only word I can say is he is a phenomenal player, apart from that an Arsenal player and I feel I shouldn’t say much more than this.’