The France international was a summer transfer target for both the Premier League duo in the summer window with offers being tabled by both clubs before the deadline.
However, Monaco resisted all offers for the 21-year-old after earlier losing Benjamin Mendy, Tiemoue Bakayoko and Bernardo Silva but he could well be on the move next summer.
"Liverpool were really interested," Vasilyev told Sky Sports. "But it was Arsenal that was 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."
Lemar could even go sooner with the winter window opening in January, although Monaco are less keen to do a deal in the New Year.
Vasilyev added: "I think it would be fair to say Thomas will have a fair chance to an exit next year. January, in football, never say never, but this is not what we’re thinking of, this is really not in our plans.
"January doesn’t seem like the right time for major changes in the team."