Sky Sports News understands City have offered Leicester in excess of £50m for Mahrez and it is believed this is their second bid for the Algerian winger during this window.
Sky sources understand £50m is well below Leicester’s valuation of Mahrez, who also handed in a transfer request in May 2017.
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Speaking earlier this month, Foxes boss Claude Puel insisted Mahrez would not be allowed to leave the club in January.
When asked if Mahrez could cost potential suitors £100m, Puel said: “Perhaps; maybe in the summer he will cost even more than £100m.
“It’s just rumours, noise about him. We can see Riyad smile, enjoying his football, enjoying to play with his team-mates.”
Mahrez submitted a transfer request to Leicester last year amid interest from Roma, but the Foxes refused to sell the 26-year-old after rejecting three bids from the Italian club – including a £31.8m offer.
Speaking at the time, Mahrez said: “Out of the huge admiration and respect I hold for Leicester City, I wanted to be totally honest and transparent with them and have therefore informed the club that I feel now is the time for me to move on.
“I had a good discussion with the chairman last summer and we agreed at that time that I would stay for another year in order to help the club as best as I could following the transition of winning the title and in the Champions League.
“However, I am fiercely ambitious and feel that now is the time to move on to a new experience.”
City have submitted their second bid for Mahrez after confirming Leroy Sane suffered ankle ligament damage during Sunday’s FA Cup fourth-round win over Cardiff.
Sane has been ruled out for up to six weeks and is set to miss the Carabao Cup final against Arsenal as well as both legs of City’s Champions League last-16 tie against Basel.