The 29-year-old came through the Palace academy before leaving in 2012 for Southampton.
He moved to Liverpool in 2015, but saw injuries greatly hamper his progression at the club, including a spell on loan at Bournemouth in the second half of the 2018/19 season.
Palace are suffering through their own injury spell at the moment, with the defence in particular looking short of options.
Palace signed Nathan Ferguson this summer after a deal in January fell through due to a knee injury that the defender is still rehabilitating from.
Hodgson revealed last week that Ferguson is in training, but admitted that he is unsure about when the 19-year-old will be able to return to first-team football.
“I don’t know how long it will take – he’s recovering from a nasty injury,” Hodgson told reporters.
“But in terms of the long-term future of the club, I think we have a big asset in our pocket there as well.
“But unfortunately I can’t really comment yet because I haven’t seen him.”
Joel Ward started last weekend’s Premier League fixture against Southampton, while Martin Kelly was drafted in for the Carabao Cup fixture against Bournemouth in midweek.
According to the Athletic, Clyne has been given two weeks to prove his fitness to Hodgson and his staff in the hope of securing a permanent contract.
Hodgson was in charge of England when Clyne earned the majority of his senior caps, and his versatility in being able to also play at left-back could come in handy.
Patrick van Aanholt remains an absentee after dislocating his shoulder against Manchester United in July, and though Tyrick Mitchell was in good form against Southampton, squad depth is a big issue at the moment at Selhurst Park.