The Express claims that the two clubs are front-runners to secure the services of the 29-year-old whose brother and agent believe he is good enough to play at some of Europe's top clubs.
Arnautovic has fuelled speculation over his future at West Ham in the last couple of days.
According to the agent of Marko Arnautovic, Everton are interested in the Austrian attacker.
Arnautovic's agent, his brother Daniel, says the Austria worldwide is considering a January exit and claims a host of top sides in England are monitoring his situation.
"He's a big part of West Ham's team but I think it's possible to tease out more of him. For that to happen, he has to play for a top team", he said.
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He added: "He's already playing in the Premier League and that's where the best players are".
"Someone like him should not be fighting against relegation but playing for the best seats and for the worldwide places".
Asked whether Marko could leave in January his brother responded: "That's possible".
The Portuguese coach knows Cahill well as the pair won the league together at Chelsea during the 2014/15 campaign when the 61-cap England worldwide played a key role in Mourinho's title-winning side.
Marko Arnautovic is a popular member of the squad and his presence lifts West Ham.
"He almost moved in the summer, he almost went to Manchester United", Parlour said on talkSPORT. He loves that club and the fans.
Arnautovic, who moved to east London in the summer of 2017 from Stoke for a then club record fee worth up to £25m, has been in outstanding form since being handed a more central role by former manager David Moyes last season, and scored 11 Premier League goals last term. "But you also get the feeling this can not be the end".