The two clubs pursued Lemar during the summer.
Arsenal had a bid worth £92m accepted on deadline day but ultimately made their move too late.
Liverpool, meanwhile, offered Monaco £80m in the final 48 hours of the window but were rebuffed.
Ahead of next month, the Ligue 1 giants have insisted Lemar will not be sold before the end of the season.
But French newspaper L’Equipe say Liverpool and Arsenal may try again before the January window shuts.
The 22-year-old winger is said to be the primary target with the two English sides still in contact.
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How Liverpool could line up with Thomas Lemar this season
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“Jurgen Klopp personally drove their pursuit of Lemar”
Following the £75m capture of Virgil van Dijk at Anfield, Liverpool are not finished for January.
Jurgen Klopp personally drove their pursuit of Lemar earlier this year and is determined to not give up.
With Philippe Coutinho’s future uncertain too amid Barcelona’s interest, the France international could replace him on Merseyside.
Similarly at Arsenal, Arsene Wenger must strengthen with at least Alexis Sanchez set to go in 2018.
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L’Equipe add Lemar has rejected all proposals put forward to him by Lemar to extend his deal beyond 2020.
He is apparently ready to take the next step in his career and head for England, with Chelsea potential suitors too after recently joining the race.
Lemar would even be open to a January transfer with the World Cup in his thoughts.
The France attacker needs to nail down a place in the first XI ahead of the tournament in Russia.