But Arnautovic's brother, Danijel, is his agent and refused to rule out a move away from east London.
The forward has admitted while on worldwide duty with Austria that he wants to be competing with the best players.
Arnautovic's brother represents the Austrian and he's confirmed he was "very, very close" to leaving for a Champions League club last summer.
"There are a lot of Premier League clubs tracking him".
The report adds that we are the favourites to sign the 29-year-old, as Jose Mourinho could be tempted into moving for the West Ham star due to Alexis Sanchez's lack of impact. "But for a player like Marko this can not be the end", he said.
Arnautovic has fuelled speculation over his future at West Ham in the last couple of days.
Arnautovic, who was instrumental in helping West Ham avoid relegation last season, has continued where he left off by netting five more goals this campaign, helping his team sit comfortably in the 13th position after a hard start to the season.
"He's a big part of West Ham's team but I think it's possible to tease out more of him".
Arnautovic was strongly linked with a move to Manchester United during the summer transfer window.
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. However, his brother's words will be of great concern to the club's hierarchy as they attempt to build a side capable of challenging at the top end of the Premier League.
Arnautovic is now away on global duty with Austria, who face Nations League ties against Bosnia-Herzegovina and Northern Ireland in the coming days. But there is still that feeling, that this can not be the end. Nasri is not eligible to play until 1 January because of a doping ban.