Standard Sport understands the club will reject all bids for one of their star players after an unnamed Chinese Super League club saw a £35million offer dismissed this week.
Privately, West Ham fear talk of a big-money move could turn Arnautovic's head and feel his agent, who is also his brother, is "trying to create deal to earn money."
When asked by Sky Sports about Arnautovic's future, Pellegrini said: "I will not talk about the price, I will not talk about the rumours.
"I am absolutely sure that a lot of teams will want to have Marko Arnautovic as a player in their team because he's a top player.
"But now, in this moment, he is here at West Ham and we will see what happens in the future but for the moment he continues with us."
When asked if he wants to keep Arnautovic beyond the end of the window, Pellegrini added: "I suppose every manager wants to keep those kinds of players."