The Eagles have netted just seven goals in the Premier League this season and are destined for another campaign in the lower echelons of the table.
They have fired a blank six times already this term, with Andros Townsend and Wilfried Zaha their joint-top scorers with just three.
Hodgson is therefore keen to add to his attack and believes Czech Republic international Schick could be the answer.
According to Calciomercato, the Selhurst Park club are among a clutch of sides who are considering a loan move when the transfer window re-opens in January.
The south Londoners have also been linked with a move for Arsenal forward Danny Welbeck, though that interest is likely to dampen following the serious ankle injury the England international suffered in the Europa League against Sporting Lisbon.
Schick is known in Italy for being surrounded by beautiful women and he could soon be bringing them to the Premier League if he were to join Crystal Palace.
He dates Hana Behounkova who regularly posts glamorous images from her travels around the world.
In addition, Schick's sister Kristyna is a model with over 100,000 followers on Instagram.
She might soon be swapping fashion parties for Selhurt Park to watch her brother in action.
Schick is down the pecking order in the Italian capital and has started just two Serie A games this season, failing to register as a result.
Following a season-long loan with Roma during the 2017-18 campaign, the 22-year-old completed a permanent move to the Stadio Olimpico last summer for a fee which could rise to €42million [£36m].
But with Edin Dzeko manager Eusebio Di Francesco's prefered option in attack, Schick's opportunities have been restricted and as a result a move away could be in the offing.
Italian sides Bologna, Parma and Udinese are all mentioned with an interest, while La Liga club Real Valladolid are also said to be keen.
Ex-England boss Hodgson has become increasingly agitated by his side's failure in front of goal but is hopeful his current squad can turn it around.
"I believe as long as you are creating those chances and you are getting in those positions the goals will come," said the Palace coach.
"What you do is make certain the players believe in themselves, they keep working at their game, and they share your belief that if you keep doing this then we will score goals."
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