Antonio Conte's side have reportedly identified Palmieri as an alternative to Juventus defender Alex Sandro, whose price tag is proving to be too costly for the west London outfit, but according to Forza Roma Jurgen Klopp's men have entered the race for the Brazilian, who can play at left-back or on the wing.
Liverpool's interest is said to have come off the back of recommendation from star striker Mohamed Salah, who swapped Rome for Merseyside last summer.
The Egypt international has been in phenomenal form since joining from the Giallorossi and has apparently advised the Reds hierarchy to raid his former club once again, though they already have a number of options which they can utilise in the left-back position in Andrew Robertson, Alberto Moreno and James Milner.
If Liverpool are serious about signing Palmieri, who is still on the way back to full fitness after suffering an anterior cruciate ligament injury, then they may have to act fast.
Chelsea are already in discussions with Roma according to the player's agent, Fernando Garcia, who believes it would be a "dream" for his client to work for a top coach in Conte, who is looking for a sufficient alternative to first-choice left-back Marcos Alonso.
Liverpool's need for a left-back may not be obvious, but manager Klopp could see fit to delve into the transfer market for a right-back amid concerns over the fitness of Nathaniel Clyne, who is struggling with an ongoing back problem.
There were suggestions that the Reds were interested in out-of-favour Barcelona full-back Aleix Vidal, whose representatives travelled to Liverpool for talks this week, but the Spaniard is now set to return his former club Sevilla in a deal believed to be worth around £8.8m.