Sanchez is in the last six months of his current Arsenal deal and is expected to move to Manchester City, either in the January transfer window or as a free transfer in the summer.
However, recent reports suggested that Jose Mourinho's side were looking to hijack a potential deal for the Chilean forward with a £25m ($34m) bid as Gunners boss Arsene Wenger seems resigned to losing his key man.
The Sao Paulo native is reportedly valued at £45m ($61.8m) by the Ligue 1 side and is already learning English in anticipation of a move to England.
With the Brazilian's imminent arrival at the Emirates Stadium, the likelihood of Sanchez moving to Old Trafford has increased as the former Barcelona man is understood to be desperate for a move this month.
While City remain favourites, the same report claims that Pep Guardiola's men are prepared to walk away from a potential deal due to the 29-year-old's financial demands.
Sanchez is reportedly looking for a £300,000-per-week ($412,000) deal along with a sign-on fee of up to £25m. In addition, Arsenal are looking for at least £35m ($48m) to part with him in the January transfer window.
Mourinho refused to rule out interest in Sanchez and most recently claimed that if there is a "fantastic opportunity" for a deal in January, United will not ignore it.
"Let's say we have three targets in the summer – a defender, a midfielder and an attacking player. If in the winter you do one of the three, then in the summer you do three minus one," Mourinho stated, as per the Mirror.
"If we do something now, we don't do it in the summer again. Our plan was not to have new players in January. We build a squad with this idea and I share this philosophy with Mr Woodward and the owners.
"But if there is a fantastic opportunity on the table, you fight for that fantastic opportunity. If it happens, it happens."