The Gunners made a late move for the France international on Thursday, with a £90 million bid reportedly accepted by the French outfit.
His potential move nearly allowed Alexis Sanchez to join Manchester City but Lemar decided to stay with Monaco due to the time constraints.
He played and scored for France in their 4-0 win over Netherlands but the 21-year-old insists he is happy to play in France, despite Arsenal's interest.
Lemar told Canal+: "I am a player of Monaco, I am very happy there.
"Now I'm going to make a good season with my club, to try to progress and move forward in my career with Monaco.
"I prepared for this game as usual. I do not lose my head, even if there are rumors. I was focused on my game, that's all that mattered to me."