Lemar is a top target for Premier League rivals Arsenal and has also been linked with Manchester United.
And now Liverpool are sniffing around too, The Telegraph report.
They report a club-record fee of €60m (£55m) could be made for the France international.
But it’s claimed Monaco don’t want to sell their star man, after already cashing in on a host of top talent this summer.
Bernardo Silva, Benjamin Mendy and Tiemoue Bakayoko have all departed and Kylian Mbappe is set to follow.
The Telegraph claim the attempt to get Lemar is not related to Philippe Coutinho.
Coutinho is wanted by Barcelona but Liverpool are refusing to sell.
Monaco reportedly told clubs interested in their players earlier this summer that only one of Bakayoko or Lemar would be sold.
Chelsea signed Bakayoko for £40m earlier this summer.
