Olivier Giroud has played in just five games for Chelsea this season, scoring a solitary goal.
He has been vocal about his frustration at a lack of game-time before, as has the French national team manager Didier Deschamps.
Deschamps is a fan of Giroud, who scored an important winning goal for France against Iceland during the international break.
“You know, Olivier suffers his situation. Before January, nothing will move,” Deschamps told the press.
“After, it will be up to him to see. I hope he will have more playing time.
“But we cannot take away his goals, his good performances. He scores, he makes others play well."
Hodgson wants a Premier League proven goalscorer to help bolster his attack this season for what could be a surprise challenge for European football.
Giroud's current deal runs out at the end of the season, so it's believed the 33-year-old would be available on the cheap (around £3m).
Chelsea boss Frank Lampard recently stated that he wanted Giroud to stay and help provide back-up to Tammy Abraham.
However, the former Arsenal striker cuts an increasingly frustrated figure and is likely to push for a move come the winter window.
Fellow Blue, Michy Batshuayi, enjoyed a successful loan spell at Selhurst Park last season, but Palace have been rebuffed in an attempt to bring the Belgian in permanently.
Lampard is reluctant to let go of important players while Chelsea are under a transfer embargo, but Giroud's place in the pecking order makes an exit more palatable.
Palace could face competition from Inter Milan for Giroud, but the player is thought to prefer staying in London, where he's played his football since 2012.
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