Benitez was unhappy with the club's business in the transfer market after failing to land their main targets despite Ashley promising him "every last penny" to improve the side.
"If I'm being totally honest, I don't expect to see Rafa Benitez at Newcastle over a sustained, long period of time," Carragher told Sky Sports.
"I think eventually something will happen with him and the owner and unfortunately for Newcastle this is not going to be a happy and long marriage.
"So enjoy it while he's here."
Carragher, who was part of the Liverpool side that won the Champions League in 2005, said Benitez will ensure Newcastle are not in the relegation fight.
"I don't think there's any chance of Newcastle going down, just because of the manager. He is very pragmatic and he'll make it very difficult," he said.
"In every game Newcastle have, you'll very rarely see them lose 4-0 or 5-0, as you might see teams in the bottom half of the table.
"I think that's where Newcastle will be, but I don't expect to see them in a relegation fight."
Newcastle, who have nine points from six games, will look to bounce back from their 1-0 defeat at Brighton and Hove Albion when they host Liverpool in the Premier League on Sunday.
(Reporting by Shrivathsa Sridhar in Bengaluru; editing by Sudipto Ganguly)
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