United fear that their interest in the striker is being used by Arsenal as a bargaining chip to get City to increase their £20 million offer.
Arsenal boss Arsene Wenger is willing to do business before the transfer window closes but only if a club meets their £35m asking price.
But City feel that is too much for a player in the last six months of his contract and who they could sign for nothing if they wait.
Sanchez will earn £13m a year and will be given a £30 million signing on fee by the Premier League leaders if he joins when his contract expires.
Sanchez remains desperate to join Pep Guardiola at City, who looked set to make a move after Gabriel Jesus picked up an injury on New Year’s Eve.
But City are ready to walk away and risk missing out completely if Arsenal don’t lower their asking price.
And United’s interest has made things more complicated but Express Sport understands they want things sealed quickly or will walk away.
United are keen on adding to their fire-power this month with Zlatan Ibrahimovic possibly leaving and Romelu Lukaku struggling to replicate his form from the start of the season.
Mourinho was coy on his interest on Friday but did say he thought Sanchez was a phenomenal player.
“I don’t think it is ethical or correct to be speaking about players of other clubs,” he said.
“If somebody speaks about one of my players, to say we are interested or not I wouldn’t be very happy.
“Sanchez is an Arsenal player. I don’t know but probably this weekend he is going to wear Arsenal colours. I don’t think it is correct to say things about Alexis Sanchez.
“At a general level, what I can say is that myself and the Manchester United board, we do not believe a lot in the January market. We don’t believe in signing a player just to do something.
“What we believe is there are some players in the football world where, if you have a chance to sign them, whether it is in January, March or July, then you have to try.
“In relation to Sanchez all I can say, which everyone else says, is that he is a phenomenal player. I feel I shouldn’t say much more than this.”