The former England manager replaced Frank de Boer in the Selhurst Park dugout on Tuesday (12 September) after the Dutchman was sacked after just 77 days in charge and is already planning to bolster his squad.
According to the Daily Mail, Hodgson could offer the 25-year-old a way out of the Emirates this winter, should Wilshere continue to have a peripheral role in Arsene Wenger's plans. The new Palace manager is understood to be a big fan of Wilshere, who he named in the England squad 29 times during his spell in charge of the Three Lions including for the 2016 European Championships, and will monitor his performances over the next four months.
In the summer, Wilshere was reportedly told he would be allowed to leave for the right offer, even though Wenger said in April he would be offered a new deal.
Like his teammates Mesut Ozil and Alexis Sanchez, the midfielder only has 12 months left on his contract and Arsenal could decide to cash in when the transfer window reopens, rather than running the risk to lose him for free next summer.
Wilshere is yet to play a game for Arsenal this season but Wenger has repeatedly stated that he will use the early stages of the Europa League to give youngsters and squad players a chance to shine and the midfielder could feature against Cologne on Thursday (14 September) evening.
Once considered to be Arsenal and England's future, Wilshere's career has been hampered by a number of injuries and fitness issues, which convinced Wenger to allow him to move to Bournemouth on loan in a bid to regain fitness.
The England international made 27 league appearances – including 22 starts – for the Cherries last term, more than in the previous two seasons combined, before suffering a broken leg that brought his spell on the south coast to a premature end.
During the summer, the Stevenage-born midfielder attracted interest from Newcastle United, Swansea, West Ham United, Turkish clubs Antalyaspor and Trabzonspor and Serie A outfit Sampdoria. However, Arsenal did not receive an offer that matched their demands, largely as most enquiries were about a loan deal rather than a permanent move instead.
Meanwhile, Hodgson said he was raring to go as he prepares to take charge of his first game as Palace manager as the Eagles host Southampton in the early kick-off on Saturday (16 September).
Palace have claimed the unwanted title of becoming the first top flight side in 93 years to lose their first four games without scoring and face back-to-back trips against the two Manchester sides and a home game against Chelsea, after this weekend's clash against the Saints.
However, the former England manager was not fazed by the pressure.
"The pressure has always been there and it'll only get stronger," Hodgson told talkSPORT on Wednesday. "The money improves every year and the thought of relegation gets more and more frightening for the teams that are going to be threatened by it, so there's no doubt the pressure has increased."