Arsenal made a £92million bid for Lemar towards the end of the summer transfer window after agreeing to let Alexis Sanchez join Manchester City.
But the deal for Lemar collapsed after Arsenal were unable to reach an agreement with Monaco.
While Liverpool were also strongly linked in the final week of the transfer window despite signing Naby Keita and Alex Oxlade-Chamberlain.
And Monaco chief Vasilyev has revealed Arsenal were the closest to signing Lemar.
“Liverpool were really interested,” Vasilyev told Sky Sports News.
“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.”
Vasilyev then hinted Lemar was unlikely to leave in January but could leave Monaco next summer.
He added: “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.
“January, in football, never say never, but this is not what we’re thinking of, this is really not in our plans.
“Then we have to wait and see, how the team does, in different competitions, if we have an interest in him, and also his ambitions after the World Cup. January doesn’t seem like the right time for major changes in the team.”