The Mirror says the Argentine goalkeeper's current deal expires in the summer. But Speroni is set to trigger an automatic extension for another 12 months.
He has made 22 Premier League outings this season as he bids to end his career at Selhurst Park.
Speroni said: "I still feel fit and have a good few years left in me yet. Not once have I ever wanted to play for any other club in England."
Palace are also keen to tie-down highly-rated defender Joel Ward.
Ward has resumed contract talks and has been assured he is a key part of Alan Pardew's future plans.
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