It was reported earlier this week that City had put an offer of around £20m on the table for the 29-year-old, but widespread reports in England now claim that United have entered the bidding, with an offer that comes in closer to Arsenal's asking price of £35m.
English publication 'The Daily Mail' reports that Jose Mourinho's United have bid around £25m for the former Barcelona star, whilst the use of Armenian playmaker Henrikh Mkhitaryan as a makeweight has also been considered.
Sanchez is reported to have agreed terms with City over a £250,000 per week deal, but it is thought that United have no problem matching or even bettering that figure.
Renowned journalist Gianluca Di Marzio alleges that Mourinho has been in personal contact with Arsenal boss Arsene Wenger as he looks to push through a shock swoop.
With Sanchez thought to be keen on teaming up with former manager Pep Guardiola once again it is unclear whether he would be tempted by a move to the red half of Manchester.