The Reds emerged as late contenders for the signature of Lemar, who was coveted by Arsenal for the majority of the summer.
It is believed that the club were preparing to increase their offer to £75m, but according to The Mirror, the Reds have shelved their interest.
The report claims that Monaco are not prepared to sell for a figure lower than £75m, and Anfield chiefs feel that they will not have enough time to get a deal over the line.
Liverpool are now said to be focusing on coming to an agreement with Arsenal over the potential capture of Oxlade-Chamberlain, who has reportedly turned down Chelsea in favour of a move to Merseyside.
It appears that negotiations are not running entirely smooth following reports that the Gunners have rejected the Reds' opening offer of £30m, with the club angling for £40m.
The England international has one year remaining on his current contract at the Emirates.