The Brazil international is keen on completing a switch to Camp Nou after he was forced to continue at Anfield despite the midfielder handing in a transfer request in the summer. The Catalan club are believed to be preparing an offer of around £140m ($189.9m) in bringing the former Inter Milan man to Barcelona.
According to the Independent, Liverpool are keen on retaining Coutinho beyond the ongoing window. However, the 25-year-old is likely to complete his move to Spain as early as next week, and this has pushed the Merseyside club to consider other options to strengthen the squad.
Lemar has been identified as a potential replacement for Coutinho, with manager Jurgen Klopp open to the idea as well.
However, Liverpool are aware it is difficult to negotiate with Monaco in convincing them to allow the French international to leave the Ligue 1 winners in January.
Liverpool's league rivals Arsenal are also believed to be interested in signing the Monaco forward. Despite all the speculation, the French club's manager Leonardo Jardim has ruled out Lemar's exit in the mid-season transfer window.
This could convince the Reds to turn their attention towards Mahrez, with the Algerian international identified as an alternative target to Lemar.
Mahrez was keen on making a move away from the King Power Stadium in the summer, and AS Roma came close to securing his services after making multiple bids to secure his services. However, the Serie A club's final bid did not match the Foxes valuation of the player.
Leicester manager Claude Puel, though, has insisted Mahrez is happy to continue with his current employers.
"I want to keep all my best players and Riyad is an important player in the team for us. When I discussed things with him, he had a good attitude and is enjoying his football," Puel explained.
"He is a happy player and likes to play with his teammates. His statistics have improved, with assists and scoring. Of course, I want him to continue all this good work.
"He is happy with the squad, it is speculation, but for me it is no problem. He is happy to play with this team, he enjoys his football and this is the most important thing."
In addition to moves for Lemar and Mahrez, Klopp is also looking to bring in a new goalkeeper in January.
The Reds are interested in signing Atletico Madrid's Jan Oblak, but the Slovenian international has a £100m release clause and is wanted by other European clubs as well. They have also monitored AS Roma's Allison and Stoke City's Jack Butland as targets for the position.
The German manager currently has the services of his compatriot Lloris Karius and Belgium international Simon Mignolet as his two first-choice keepers.
While defender Virgil van Dijk became Liverpool's first signing of the winter transfer market, the Anfield club remain hopeful about signing an attacker and a stopper to strengthen the squad.