The 28-year-old is into the final year of his contract with the Gunners and has made it clear that he does not want to sign a new deal with the side as he seeks a significant increase on his £130,000-a-week wages.
City were in hot pursuit of the Chilean over the summer and reportedly had a bid of £75m rejected in the final throes of the transfer window as Arsene Wenger's side had failed to secure a replacement.
Arsenal now face the prospect of either having to accept a significantly-reduced bid in January or lose Sanchez for free next summer, and according to The Sun, that situation has piqued the interest of United boss Jose Mourinho.
The newspaper claims that United have been quoted a signing-on fee of £25m should they wish to tempt him to Old Trafford at the end of the season, while a pre-contract could be signed when the transfer window re-opens in January.
Sanchez is expected to feature for the Gunners tonight when they welcome League One side Doncaster Rovers in the EFL Cup.