The Arsenal forward looked set to be reunited with Pep Guardiola, whom he played under at Barcelona, only for his £60m (€67.6m, $81m) move to collapse on deadline day after the Gunners failed to secure a replacement.
However, the 28-year-old has turned down an offer to extend his deal at the Emirates and Arsenal are resigned to lose him for free next summer.
While Sanchez's move to City did not materialise, it was widely acknowledged he would complete the transfer at the end of the season but United have reportedly entered the race for the former Udinese and Barcelona star.
According to The Sun, the player's representatives sounded out United after speaking to their cross-town counterparts when it became apparent that Sanchez wanted to leave Arsenal and Mourinho has reportedly been quoted a £25m signing fee for the player.
While Arsene Wenger is reluctant to sell the Chilean in the January transfer window as he feels parting way with Sanchez would undermine his side's chances of securing a top-four finish, the striker will be allowed to speak with foreign clubs from 1 January.
Sanchez would not, however, be able to sign a pre-contract with the 20-time champions of England as it would contravene the rules. An agreement could still be put in place for the deal to go through when his contract at the Emirates expires next summer, however.
City are still expected to be in pole position to land Sanchez but the blossoming partnership between Sergio Aguero and Gabriel Jesus could see Guardiola revise his plans. The former Barcelona and Bayern Munich manager had made Sanchez his number one target in the summer, despite the fact that fitting the Chilean in the team would have in all likelihood meant jettonising Aguero.
Having started on the bench against Chelsea on Sunday (17 September), Sanchez could feature as Arsenal host Doncaster Rovers in the third round of the Carabao Cup on Wednesday night.
However, Danny Welbeck will not be available for selection as he faces another lengthy spell on the sidelines after picking up a groin injury at Stamford Bridge.
The 26-year-old has been plagued by injuries in recent campaigns and only returned from a long-term knee injury at the back end of last season but made a fine start to the campaign and is Arsenal's top scorer in the Premier League with three goals from five games.