The Magpies manager has been offered an extension to his current deal, which expires on June 30.
But despite briefly chatting to managing director Lee Charnley at the club's training ground on Monday, Benitez has given no indication as to whether he is going to sign new terms, says The Sun.
And while Newcastle chiefs are still hoping the Spaniard stays, they are increasingly worried he will walk away at the end of the month and leave them starting pre-season without a boss.
Charnley has purposely not approached other managers with Benitez still under contract.
But he will soon be forced to look at alternatives with just 18 days to go before the 2005 Champions League-winning chief can leave.