According to ESPN FC's Peter O'Rourke, the news comes from an unnamed source, who reported the Slovakian "could consider moving on" if he remains behind Dejan Lovren and Mamadou Sakho in the pecking order.
Further, the source also revealed there are clubs both in the Premier League and abroad that are "actively monitoring his situation" ahead of the transfer window this summer.
The 31-year-old is in his ninth season at the club, and throughout his time at Liverpool, he has often been one of their more reliable options in central defence. But he has endured a poor campaign this year.
Indeed, after returning to fitness in late February following two months out injured, he has been unable to force his way back into Jurgen Klopp's plans and has put in several subpar performances in the appearances he has made.
Bleacher Report UK's Jack Lusby was one of many to criticise Skrtel for his part in Liverpool's 3-2 defeat to Southampton, in which the Reds were two goals ahead prior to him coming off the bench and subsequently lost after he came on:
What has complicated the situation for Skrtel is Lovren's improvement under Klopp from arguably Liverpool's worst central defender to their best. He even grabbed the winning goal in their electrifying UEFA Europa League match with Borussia Dortmund on Thursday, per BT Sport's Jake Humphrey:
Because of Skrtel's own poor form, he is not only behind Lovren and Sakho in central defence but also Kolo Toure and midfielder Lucas Leiva in Klopp's thoughts, with the pair starting ahead of him against Bournemouth on Sunday, per Gianluca Di Marzio's Joseph Musker:
The Reds are likely to pursue defensive reinforcements this summer—they are on course to match last season's disappointing Premier League goals-against tally of 48 and have kept just two clean sheets in their last 10 league games—so Skrtel may find his position in even more jeopardy if he does not seek pastures new.
One player who has enjoyed a new lease of life at Anfield in recent times is Joe Allen, and according to the Mirror's Hamish Mackay, Klopp has given a strong hint the Welshman is in line to earn a new deal at Liverpool to extend his stay beyond the end of next season.
Klopp said: "He deserves a future here, of course. He is in brilliant shape. We will see what happens but for me it is not the time to talk about his contract now."
Allen stood out for the Reds in their 2-1 win over Bournemouth on Sunday, as Liverpool revealed:
Squawka Football gave further insight into his impressive performance:
As noted by Musker and writer Dave O'Connell, the midfielder is getting better and better with each game:
If he can maintain his renaissance, he can become an incredibly useful player for Liverpool, for whom central midfield has been something of a problem position in recent years with Steven Gerrard's decline and departure and Jordan Henderson's initial struggles following his arrival.
Allen has only just turned 26 and is entering the peak years of his career, so a contract to ward off any potential interest in him this summer would both be deserved and a strong piece of business.