The Gunners were reported to have bid £92 million for the France international on Thursday in a deal that would have paved the way for Alexis Sanchez to join Manchester City.
But Arsenal could not get the transfer over the line and Wenger has now revealed that it was Lemar's decision to remain in the principality.
"The player has chosen to stay in Monaco," Wenger told beIN Sports.
"You do not want to talk too much about... you have to respect what happens on the negotiation side.
"I know we live in the era of transparency but at some stage, to respect everybody you have to not talk too much about that."
Arsenal subsequently refused to sell Alexis, who has less than one year remaining on his contract and has indicated he will not extend his stay in north London.
Still only 21, Lemar joined Monaco for a small fee from Caen in 2015.
He scored 14 goals in all competitions last season, his second at the club, and played a significant role as they won the French championship and reached the Champions League semi-finals.
Monaco subsequently fielded interest in several players on the back of that campaign, with Tiemoue Bakayoko, Benjamin Mendy, Bernardo Silva and Kylian Mbappe all departing.
But Lemar has decided to stay and help Leonardo Jardim's team defend their Ligue 1 title, stating he is "very happy" at the club.