The long-serving goalkeeper, who has spent the past 10 years at Selhurst Park, is approaching the end of his current agreement.
If no fresh terms are put in place, Speroni will drop into the free agent pool in June.
Palace are eager to avoid such an occurrence, with management and supporters keen to keep the Argentine on board.
Co-chairman Steve Parish is, however, aware that a meeting may have to be delayed as everyone connected with Palace looks to enjoy a well-earned rest at the end of a productive first season back in the Premier League.
He told the Croydon Advertiser: "There is no date set but I think everyone wants a little break.
"Tony (Pulis) is going to have a chat with Julian and talk to him about where he sees him and how he sees it going forward.
"We would all like to keep Julian at the club, of course we would. It's looking for a long-term plan as well.
"I think we would like to see Julian really staying at the club and having a future, maybe on the coaching side.
"He's a massive part of the fabric of Palace.
"Right now Julian wants to play and so he should, he's a fantastic keeper and a brilliant servant for the club.
"He needs to sit down with Tony in the first instance and then I'm sure I'll get involved from there."