Chelsea have been linked with a number of strikers in recent weeks ahead of the winter window.
Olivier Giroud is yet to score this season, whereas Alvaro Morata has just three goals in all competitions.
Chelsea are unbeaten in the Premier League under Sarri and many pundits have claimed they are a goalscoring striker away from being serious contenders for the title this term.
Arnautovic has enjoyed life as a centre-forward at West Ham since David Moyes started playing him there midway through last season.
The Austrian has four goals in his eight Premier League appearances so far under Manuel Pellegrini.
Liverpool legend Steve Nicol believes the 29-year-old would be a fantastic signing for Chelsea.
But he says West Ham would find it hard to stay in the Premier League if they sold their star asset.
“It would make absolute sense for Chelsea,” Nicol told ESPN FC.
“It would make absolutely no sense for West Ham.
“If they’re to stay in the Premier League, because they look as though they’re going to be in a dogfight for most of the year, then this guy has to stay.
“If they want to stay in the Premier League, West Ham, he has to stay.
“But, if you’re Chelsea, you have to go and get him.
“I think he would fit in fantastic.
“His attitude, he’s got the kind of attitude that’s infectious.
“People say he’s moody, but I’ll tell you what, I’ll have a moody guy that scores goals and upsets the opposition any day over Giroud or a Morata, who I wouldn’t be that bothered about playing against.
“This is not going to happen, but it would make sense for Chelsea.”