Speculation emerged earlier this week that the Red Devils are prepared to rival Manchester City for the Chile international, who has less than six months left to run on his current Emirates Stadium deal.
Sanchez, close to becoming a Man City player last summer in a £55m move, is said to be weighing up his options as he allows his contract to tick down.
In the wake of media reports linking United with the 29-year-old, however, Mourinho refused to give too much away.
"I don't think it is ethical or correct to be speaking about players of other clubs," he told reporters. "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 for sure 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 with 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 then you have to try. All I can say is that Sanchez is a phenomenal player."
Sanchez has scored eight goals in all competitions for Arsenal in 2017-18 and has set up three more for his teammates.