The England shot-stopper only has a further 19 months to run on his existing deal, with Liverpool among those reportedly interested in the 27-year-old.
Nevertheless, Pellegrini, who dropped Hart for a number of games last season, says that negotiations are already underway over a proposed extension.
"I think it's an issue that we're talking about," the Chilean told Sky Sports News. "I think in the right moment, [we will] have the right solution. I think Joe wants to stay here and we want him to stay."
Hart has managed two clean sheets in seven Premier League outings so far this season.