The 72-year-old has been a revelation for Palace, securing their future in the Premier League.
But he could be prepared to walk away if he is not given assurances about transfers.
The Times suggests that Hodgson wants to overhaul the squad.
If he is to stay at Palace beyond this summer, he wants guarantees that he will have money to spend on players.
He also wants to know funds raised from selling the likes of Wilfried Zaha would come back to his transfer kitty.
If the club and Hodsgon find an agreement, he would sign a one year extension to his deal.