Sky Sports is reporting that the Red Devils are considering a potential bid to sign the Austria international in the 2018 summer transfer window.
The same article states that West Ham would be reluctant to sell the 29-year-old following his impressive first season at the east London outfit.
According to the same story, Arnautovic is valued at around £50m despite West Ham signing the playmaker in a club record fee of £25m last summer.
The Hammers star netted 11 goals and made six assists in the 2017-18 season to help West Ham avoid the drop under former Manchester United boss David Moyes.
Arnautovic, who has four years left on his current West Ham deal to ensure the Hammers can ask for a premium price, worked with Jose Mourinho during a brief stint at Serie A side Inter Milan.
The Austrian forward netted 26 times in 145 games in all competitions during four seasons at Stoke City.
Manchester United were 1-0 winners against Chelsea FC in the FA Cup final at Wembley after Eden Hazard scored the winner from the penalty spot.
The Red Devils finished in second place in the Premier League table but 19 points behind champions Manchester City.
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