Austria international Arnautovic only joined the Hammers last summer but is already being linked with a move away after impressing for the east London club.
United have signed two players so far this summer – with Diogo Dolot following Fred through the doors at Old Trafford – but the club are being linked with a number of other moves.
Mourinho signed Romelu Lukaku last summer and Alexis Sanchez in January but he remains interested in strengthening his attacking options.
And the Portuguese manager is said to have expressed interest in signing Arnautovic, who he briefly worked with at Inter Milan.
According to Austria outlet Osterreich, Mourinho has shared text messages with Arnautovics brother and agent, Danijel.
Last week, Arnautovic was asked about the interest from Manchester United and admitted he was flattered to be linked with the Premier League giants.
Its nice to hear, I havent heard anything from my brother, my agent, he said.
You can keep asking this question but youll never hear the answer from me.
Arnautovic scored 12 goals in 38 games in all competitions last season and has the potential to succeed at United, according to Stuart Pearce.
The former England defender told talkSPORT: He has got the ability to be in the top four in this country without a doubt.
Its just whether West Ham want to sell him.