According to the United manager, he isn't necessarily confident of striking a deal but believes they have a chance if the Chilean does want to leave.
"I am not confident, but also not unconfident," said Mourinho. "Just relaxed and we have the feeling that he is an Arsenal player. We have the feeling that he can stay there.
"But also the feeling that he can move and if he moves, I think we have a chance. But, I think most probably, a player like him has other big clubs interested. Who knows?
"I think him, Mr Wenger and Mr Gazidis (Arsenal's chief executive), I think they know really what is going to happen. I have no idea to be honest.
"I'm not going to speak about hypothetical situations, he is an Arsenal player and he can be an Arsenal player. But I feel I shouldn't speak about him because he is an Arsenal player."
Signing Sanchez would be a huge coup for United. He's 29-years-old, world class and only has six months remaining on his Arsenal contract, meaning he'll come cheap.
Whereas the Chile international was valued at £60 million by Manchester City last summer, he's now being linked with a £30 million move to the Theatre of Dreams.
United fans are understandably excited about the idea of signing Sanchez, especially because he's coming from Arsenal, but there's reason to be concerned by his imminent arrival.
Sanchez will reportedly be made United's highest-paid player when he puts pen to paper with wages of £350,000-per-week.
And it's for that exact reason Bayern Munich rejected the chance to sign him last summer.
German football expert Raphael Honigstein claims Bayern said no to signing Sanchez in 2017 out of fear his ridiculous wage demands would cause trouble in the dressing room.
Bayern needed to make Sanchez their top earner to sign him, which wouldn't have gone down well with star players like Robert Lewandowski and Thomas Muller.
It was a smart move from the Germans and one that Mourinho should bear in mind. There's every chance Sanchez's wages could cause disruption in United's dressing room.
If the Arsenal man is worth £350,000-per-week, players like Romelu Lukaku, Anthony Martial and Paul Pogba may question what they're worth.
Sanchez has previously caused divides in dressing rooms, too, so history could repeat itself if Mourinho isn't careful.
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