The Eagles have been in dismal form in recent weeks having not won any of their last eight Premier League games.
It has left them 15th in the table and there is a fear that the south London club are sleepwalking into a relegation battle.
Hodgson's contract expires at the end of the season but Palace might terminate the 76-year-old's deal now to arrest the slump.
And Cooper is one of the top names being mentioned by Selhurst Park chiefs as a replacement, talkSPORT understands.
While Palace would have to pay Hodgson off to sack him, they would not need to spend a penny on a compensation package for Cooper after he was sacked by Nottingham Forest earlier this month.
The Welsh coach has been on Palace's managerial shortlist for some time and had already been earmarked as a potential successor when Hodgson's contract was up next year.
The fact that Palace wouldn't be required to pay any compensation to get free agent Cooper could see them strike soon. However, whether Cooper would prefer a short break away from management remains to be seen.
The former Swansea boss has also been linked with Birmingham as a possible Wayne Rooney replacement, who has had a torrid two months in charge.
But talkSPORT understands that this remains unlikely at this point in time.
Most read in Football
Roberto Firmino faces uncertain Saudi future amid Premier League interest
Arsenal told to be ruthless and sell academy star to fund Premier League title bid
Messi shows the grind never stops with ripped post-workout photo alongside wife
How will the Africa Cup of Nations affect the Premier League this season?
Palace are also said to be keeping tabs on Kieran McKenna after an incredible stint with Ipswich Town, who are gunning in on back-to-back promotions.
The Eagles have won juts twice in their last 16 games, their latest outing a 2-1 defeat to Chelsea on Wednesday.
It has left them three points ahead of Luton Town, who are 18th in the table but have seen an upturn in form with wins over Newcastle, Sheffield United and Palace recently.
Hodgson was asked about his future after the defeat to Chelsea, saying: "The next question is whether I'm frightened to death now because we're only a few points above the relegation zone. I'll answer that one for you as well: No, I'm not.
Steve Cooper serenaded by Nottingham Forest supporters with manager on the brink of City Ground sacking
"It doesn't matter to me who is top of the league at this moment in time.
"It didn't matter to me that Aston Villa could have won last night and gone top or lose and finish third or fourth.
"What bothers me is 38 games and where we're going to be.
"And what I think is that with this team and the way we've played in the last four games, home to Liverpool where we didn't deserve to lose that one.
"Manchester City where we defended well and got ourselves a credible point. Brighton, tough team, 1-1 and a point."