The Austrian has been linked with a move to Old Trafford, and former Potters team-mate Crouch wrote in a column for the Daily Mail (h/t Goal's Chris Burton) he always believed such a move could happen:
"Marko Arnautovic always struck me as a player who could end up at Manchester United.
"Old Trafford looked like it would be the perfect stage for him.
"Some of you may laugh at that statement but, believe me, Marko has always had special ability.
"There were doubts about his temperament and maybe that's why one of the top six haven't taken him.
"But the next best thing for him was a club such as West Ham, on the up in the Premier League.
"They made him their main man, something he always wanted to be, and he has thrived."
Crouch also said a move to China would come too soon for the 29-year-old, who is playing some of the best football of his life in east London.
Per Burton, Arnautovic has been linked with Chelsea and the Chinese Super League. An unnamed club from China has reportedly bid £35 million for the forward, and his brother urged the Hammers to cash in on Thursday, per Ed Aarons and Jacob Steinberg of the Guardian.
West Ham released a statement saying Arnautovic is not for sale, but sportswriter Oliver Kay wasn't sure their stance on the matter was the right one:
Oliver Kay @OliverKayTimes
I like it when clubs stand firm in these situations, but if -- IF -- West Ham are smart enough, they'll be looking at whether it might make sense to cash in Arnautovic (talented but 29, expensive, inconsistent and now unsettled) and find a better, younger player to replace him. https://t.co/FuxRWbh4Zv
Arnautovic has already scored seven goals this Premier League season, just four shy of his best-ever return in a single campaign. He's been a revelation since joining the Hammers after previously struggling for consistency with Stoke.
Described as "mercurial" by Burton, sportswriter Seb Stafford-Bloor questioned the wisdom of keeping him around if his mind is set on a big-money move:
Oliver Holt expects him to get his wish:
The switch to China appears the most likely at this point, as clubs from the Far East will not be afraid to spend big on another star. United are in a period of flux, with interim manager Ole Gunnar Solskjaer pulling the strings, and they're expected to reserve their biggest spending until the summer.
Arnautovic could be a fantastic mid-season impact signing, but the Red Devils have turned to their younger players and are building for the future. The Austria international does not appear to be a fit for the team's current state.