Arsenal launched a £90m (€100m) bid to try and sign 21-year-old Lemar on the final day of the summer transfer window as they looked to bring in a replacement for Alexis Sanchez, then on the brink of joining Manchester City. Both deals fell through however, with Liverpool also finishing the window disappointed after seeing their own efforts to sign the France international come up short.
Jurgen Klopp's side emerged as a serious contender to sign Lemar late on, making two bids in the final week of the window with the club's medical team reportedly left on standby during that time to travel to Paris to carry out a medical if a deal was struck.
Speaking to Sky Sports News, Vasilyev explained it was Arsenal who came the closest to striking a deal ahead of their Premier League rivals, but offered both clubs hope of clinching a deal next summer.
"Liverpool were really interested," Vasilyev said. "But it was Arsenal that were really close to finalising the deal.
"It was quite close. At the same time it was just too late, not enough time, to really follow up on this deal. Not all the elements of the puzzle fell in place. 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. We had some players who were supposed to leave this summer but didn't leave because of X, Y, Z. I think it would be fair to say Thomas will have a fair chance to an exit next year."
Monaco won't rule out the possibility of striking a deal with either Arsenal or Liverpool when the transfer window reopens in January, but suggest it would take an incredibly tempting offer to coax them into during business mid-season.
"January, in football, never say never, but this is not what we're thinking of, this is really not in our plans."