The 26-year-old has become the leading contender to replace Philippe Coutinho at Anfield after his £142m move to Barcelona last week.
But despite having previously handed in a transfer request and revealed a desire to play in the Champions League, Mahrez is not keen on a move to Liverpool.
According to The Express he would much prefer to link-up with Arsenal, who he almost joined in the summer.
But any switch to The Emirates Stadium is dependent on Arsenal selling Alexis Sanchez.
Having lost Coutinho to Barcelona, there is nothing preventing Liverpool from making a move for Mahrez, who is valued at £50m by Leicester [The Mirror].
Jurgen Klopp has already acquired Virgil van Dijk from Southampton but there is still money remaining for Liverpool to make a significant signing.
Mahrez submitted a transfer request in the summer in order to force through a move to a top flight European club, but moves to Arsenal, Roma and Barca collapsed over Leicester's financial demands.
After a lean campaign last season, Mahrez has enjoyed a fine return to form by scoring or assisting 14 goals in his last 17 outing in all competitions.
Leicester are ready to back Claude Puel in the January transfer window and are pledging to keep all of his key players and the French boss expects Mahrez to remain in the East Midlands.
"I can only say the same thing - there is nothing new about him," Puel told reporters, according to Goal.com. "He is a valuable player for us and I want to keep him.
"I have no problem with him. He works hard in training and is enjoying his football and he is playing very well. After that, it is just speculation. Last week, for example, he was said to be signing for another club - and all this speculation.
"I am happy he is staying here and he has a super attitude. If you saw the training session this morning he was smiling and enjoying it. It is amazing."