Darmian struggled to make a major impression on Jose Mourinho when the Special One arrived at Old Trafford last summer.
But the Italian full-back did gain more game-time towards the end of last season and produced a string of encouraging displays.
The 27-year-old, who has two years left on his current Red Devils deal, has been linked with leaving the club for several months.
A number of clubs in his homeland are known to be keen but it seems he could now stay put in England.
Italian outlet Transfermarketweb say Mourinho is no longer willing to let him leave over the coming weeks and wants to keep the defender for next term.
And it is said that United have turned down Inter Milan, AC Milan, Roma and Napoli as they seek to hold on to their ace.
Last week, Darmian remained coy over his United future following Italy's win over Uruguay.
He played the full 90 minutes as the Azzurri ran out 3-0 winners over the South Americans.
Speaking to reporters after the match, he said: "Surely this is not the moment to talk about the transfer market.
"I wouldn't know what to answer. I have a contract with Manchester United, so let's see what happens."
Darmian, who has won 29 caps for Italy, joined United from Torino for £12.7m in July 2015 and has made 68 appearances for the club, scoring once.
