Both Premier League clubs have been tracking the French winger since last summer. Arsenal offered £90m ($126.4m) for his services as a potential Alexis Sanchez replacement in August only for a deal to not materialise.
Meanwhile, Liverpool are reported to have earmarked the 22-year-old as a replacement for Philippe Coutinho who departed Anfield earlier this month for Barcelona.
The need for an attacking reinforcement may be even greater for the Reds with the outgoing loan deal for Daniel Sturridge along with recent results that saw them lose 1-0 to bottom-placed Swansea City and 3-2 to West Bromwich Albion in the FA Cup fourth round.
However, Vasilyev claims Lemar will be remaining in the south of France to help the club achieve its goals this season. He adds that it will remain the case even if they receive a big offer before deadline day on Wednesday (31 January).
"He will stay at Monaco until the end of the season at 100 per cent," he told RMC via Goal.com. "If Liverpool or Arsenal come with a big offer before the end of the market? This is not relevant.
"We will say no because the objectives are clear. Without important players like Lemar, we will not succeed."
Both clubs will face an uphill battle to land Lemar in the summer as well as it was recently reported that the French international only had eyes for a summer move to Barcelona.
Despite the arrivals of Coutinho and Ousmane Dembele this season, the La Liga leaders are still interested in adding to their squad with Lemar being on their wishlist as well.
Lemar has scored two goals and provided five assists in 16 league games for the reigning Ligue 1 champions this season.