Per Simon Stone of BBC Sport, Mourinho was relatively coy when he was quizzed a possible January move for the 29-year-old:
According to Mark Ogden of ESPN FC, United have been handed a boost in their pursuit of the player, as rivals Manchester City are unwilling to pay more than £20 million for Sanchez, whose contract runs out at the end of the season.
It's suggested the Gunners want £35 million for Sanchez, while the player's agent fees would reportedly cost around £5 million.
Mourinho added that "we believe that there are some players that if you have a chance to sign them, no matter the month, then you have to try," per football writer Chris Winterburn.
The manager's refusal to shut down the ongoing speculation suggests United's interest in Sanchez is serious.
However, according to Dean Jones, City remain admirers of the Arsenal man, and they are his preferred destination still:
Per Ogden, City actually came close to signing Sanchez at the end of the previous transfer window, although a £60 million move fell through when the Gunners failed to land an adequate replacement.
As the sharpshooter has not signed a new Arsenal contract, the Gunners now face losing Sanchez for below his market value midseason or for nothing at the end of the campaign.
City have appeared to be in the box seat for a long time in the pursuit of the player, although now the Red Devils look as though they're neck and neck with their local rivals.
According to Samuel Luckhurst of the Manchester Evening News, there is a financial factor that Sanchez is considering:
If he was to move to City, the Arsenal No. 7 would be linking up with a tremendous coach in Pep Guardiola; the pair worked together during a stint at Barcelona, too. Additionally, he'd be almost certain to earn a Premier League winners medal, with City 15 points clear at the top.
United are also an appealing proposition. Sanchez would become a much more important player at Old Trafford than at the Etihad Stadium, while the determination to get him in January would also surely count for something.
James Robson of the Manchester Evening News believes it's tough to say where the Chilean will end up:
From an Arsenal perspective, Gunnerblog's James McNicholas was demoralised by the situation:
Whatever side does end up with Sanchez will be getting one of the best players in the Premier League, as he is a serial creator and scorer of chances.
It's why United may be tempted to put up the money for Sanchez now despite his contract situation. Given he'd be able to help the Red Devils secure a spot in the top four and potentially challenge for the UEFA Champions League, the aforementioned valuation may be a price worth paying.