The Austrian striker has started the season strong scoring two goals, despite the Hammers having not collected a single point.
And Bellamy thinks Arnautovic could become restless if the east London club continue to struggle.
"I've always been aware of his ability but if he had one good game then he'd go missing for the next three or four," the former Wales international said.
"When he was out wide if he had a lot of the ball then brilliant but then sometimes he'd lose interest. He frustrated because I knew there was a real player there but at times he would just look disinterested.
"Credit to David Moyes, he put him up top and he's involved in the game all the time. I've been so impressed with him up front. He's the one player for me at this present moment that West Ham do have with a bit of pace and do have as a threat.
"However, he's a volatile player as well. If it isn't going his way come January and West Ham are struggling, he's one player, if there is going to be interest, that's going to test you."