The France international's impressive displays garnered interest in the previous transfer window with Vasilyev saying Premier League sides Liverpool and Arsenal were keen to sign Lemar and would have another chance to do so next year.
"Liverpool were really interested," Vasilyev told Sky Sports. "But it was Arsenal that was really close to finalising the deal.
"It was quite close... I'm very happy he stayed with us. For us, we need a player like Thomas Lemar. He's very important to our team.
"Definitely next summer will be something to look at, although you know in football there are no guarantees... I think it would be fair to say Thomas will have a fair chance to an exit next year."
Vasilyev said Lemar's chances of leaving in the mid-season transfer window in January were minimal as it was not the right time to make major changes to the team.
The 52-year-old also said Monaco were considering selling defender Fabinho in the next window after receiving interest for the Brazilian from Premier League leaders Manchester City and Manchester United.
Monaco trail leaders PSG by three points after eight games in Ligue 1.
(Reporting by Aditi Prakash in Bengaluru)
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