United are said to have made a £25m (€282.m) bid for Sanchez on Thursday (11 January), with reports claiming Mourinho is confident his side can beat neighbours Manchester City to one of the biggest deals of the January transfer window.
City, who saw a deadline day move for the 29-year-old fall through last summer, have tabled a £20m offer for the former Barcelona and Udinese forward but are not prepared to increase their offer – with Arsenal demanding £35m for a player who has less than six months remaining on his contract.
Reports of City's decision to abandon their pursuit of Sanchez unless Arsenal and the player himself lower their demands emerged hours before Mourinho addressed the press ahead of Monday's Premier League clash with Stoke City.
And while the United boss was not drawn on reports of their bid for the player, he admitted the club have a duty to be in the market for that sort of calibre of footballer when they become available.
"If somebody speaks of one of my players to tell we are interested or we are not interested, I wouldn't be very, very happy," Mourinho told reporters, the Manchester Evening News report.
"Sanchez is an Arsenal player, I don't know but probably this weekend he is going to defend Arsenal colours so I don't think it's correct to say things about Alexis Sanchez.
"At the general level what I can say is myself and the Manchester United board, the owners, we don't believe a lot in the January market, we don't believe in signing a player just to sign a player, just to do something, we don't believe in that.
"What we believe really is there are some players in football world if you have the chance to sign them in January, in March or in July, you have to try and that's it.
"But in relation to Alexis Sanchez I don't say a word, the only word I can say is he is a phenomenal player, apart from that an Arsenal player and I feel I shouldn't say much more than this."