Arnautovic wants to quit West Ham for a big-money move to Shanghai SIPG, who are believed to be offering to almost treble his wages.
West Ham have turned down an initial bid and maintain they do not want to sell their joint top-scorer.
But if a satisfactory offer for the 29-year-old is received, Hammers boss Pellegrini will not stand in his way.
"If I say the player must stay, the player will stay," he said. "I don't have any problem with the club. In all the months I've been working here I never had any problems with the owners about that.
"So they are not forcing me to sell Marko Arnautovic. But I think as a manager the best thing is to take the solution that is best for the player and the club, both together.
"If they agree, we don't have any problem to change.
"I am not telling the club about it. If it's a good deal for both parties we will arrange to be without Marko. But the club doesn't want to sell him."
Two seasons ago Dimitri Payet, another fans' favourite at the London Stadium, refused to play until he got a move back to his former club Marseille.
Pellegrini says he does not blame Arnautovic for having his head turned but is convinced the Austrian forward will continue to do his best for the team if the move does not come off.
"Marko is working with the team without any problem," said Pellegrini.
"When you have such a high offer, everyone wants to leave, but he has a contract here and we will know what will happen about it in the coming days.
"Of course you can't blame him, he knows what is the best option for him and now we will see what happens between the clubs. I don't think that the player will refuse to play."