Cornet has featured very little for West Ham United this season, falling down the pecking order behind the likes of Mohammad Kudus, Said Benrahma and Pablo Fornals.
The £17.5m-signing has made just two Premier League appearances for the Hammers under David Moyes this season, one of which was as a 94th-minute substitute on the opening weekend.
Cornet did start in their draw at Sheffield United just over a week ago, given the injuries to Lucas Paqueta and Michail Antonio, with Kudus also away at the Africa Cup of Nations, and he found the net to open the scoring.
However, with Kudus now back from the tournament and Paqueta closing in on a return, West Ham appear willing to let the player leave before the deadline.
The winger's lack of involvement at the London Stadium saw him snubbed by Ivory Coast for their AFCON squad, as West Ham's Europa League campaign only saw Cornet utilised as a substitute infrequently.
Nottingham Forest and Everton have both been linked with the player, but Crystal Palace have emerged as the likely candidates to land the 27-year-old.
Everton are struggling to gather the funds together to make a move for any players, while Forest have dropped out of the running after the signing of Giovanni Reyna from Borussia Dortmund.
However, the Daily Mail claim that Palace boss Hodgson is having second thoughts about the deal, despite the club making an enquiry for the player.
Hodgson has seen Jordan Ayew return from the Africa Cup of Nations this week, and the manager was also able to welcome Michael Olise back from injury against Sheffield United on Tuesday.
Olise put on a stellar display with a goal and two assists, and the work Ayew does for the team does not go unnoticed by Hodgson, so there may not be room in the side to accommodate Cornet.
Palace have already been active in the market, signing right-back Daniel Munoz from Genk, while Adam Wharton is set to follow him through the door at Selhurst Park today.
Munoz is currently injured, but Wharton and any other possible deadline day arrivals will be able to feature in the Eagles' clash with Brighton & Hove Albion on Saturday.