Arnautovic has been linked with Chelsea and Manchester United, while there have been reports of a £35 million ($44.8m) offer from the Chinese Super League that prompted West Ham to release a statement this week insisting the Austria international "is not for sale".
With the Hammers a goal up against Arsenal at London Stadium on Saturday, Arnautovic was replaced after 70 minutes and removed his gloves to wave to all corners of the ground.
And Pellegrini said the club and Arnautovic must work to find a solution that works for all parties.
"We'll see what happens in the next days," Pellegrini told Sky Sports. "It's an issue that the club must see what is the best for him, for the player, for the club, for everyone.
"I hope he will take the best decision for everyone.
"I have been in these situations a lot of times. I have been a manager for about 30 years.
"Big teams want to have the best players but I think we must remain calm about the situation.
"There is nothing 100 per cent sure so we will see in the next days how it develops."
Arnautovic joined West Ham from Stoke in 2017 and has scored seven goals in 16 Premier League appearances this season.