Cole backs West Ham to keep Arnautovic
The 29-year-old has been linked with a move away from the club in the January transfer window and Chelsea are rumoured to be interested in signing him, along with Everton.
Arnautovic has scored five goals in 10 Premier League appearances for Manuel Pellegrini's side this season and Cole said his form has helped to build a strong bond with the club's fans.
"He's happy at this club," Cole told Sky Sports.
Arnie provided a last-minute assist as Austria snatched a dramatic 2-1 victory in Northern Ireland! https://t.co/VqyQovXTaF— West Ham United (@WestHamUtd) November 18, 2018
"I don't think he's looking actively to go and leave. He's loved here.
"I don't see any reason why he should be leaving the club. Everyone knows what he's capable of here and I think we'll see him stay.
"If he does go, it will be a great loss to West Ham, but we always find a way to carry on. We always do.
"I don't want him to leave. I don't think the club will let him leave, because he's our main player."
Meanwhile, Cole gave his backing to a potential West Ham swoop for former Manchester City midfielder Samir Nasri, who is reportedly training with the club as he attempts to win a contract.
Nasri received an 18-month ban for breaching World Anti-Doping Agency (WADA) rules in 2016 but is free to train with any club again this month.
The former France international, 31, played under Pellegrini at City, and Cole said: "I think the manager has a good relationship with him.
"We know his class, he's a classy player.
"He will bring something different to the squad and he has that experience too.
"It will be a great addition to the squad but it depends what the manager wants and what the owners want."