Arnautovic, 29, is joint-top scorer for the Hammers this season with eight goals in 17 games in all competitions.
He moved to London Stadium for a club-record £20m fee from Stoke in July 2017 and his contract runs until 2022.
A West Ham spokesperson said: “Marko has a contract and we fully expect him to honour it. He is not for sale.”
The unnamed club, believed to be Chinese champions Shanghai SIPG, is understood to have bid £35m for Arnautovic with wages understood to be £200,000 a week over a four-year contract.
At a news conference on Thursday, Hammers boss Manuel Pellegrini said they had received no firm bid, but did admit it could be difficult to keep the forward in the future.
“Those are all rumours,” said the Chilean. “We know we have a very good player in Marko, a lot of clubs may want him to play for them. At the moment we have no news about him.
“I don’t want to talk about things that might happen. Marko in this moment is a player of our team, players of his quality and other players like Declan Rice, Felipe Anderson, other clubs will want them but that is one thing. I don’t think they will go.
“I want to keep him, of course, but you never know what might happen in the future.”
“Marko and all the other players involved in rumours have their heads here,” added Pellegrini.
“They want to continue playing the way they have been so far and I am sure he will have his head in the game against Arsenal.”
More to follow.