El Shaarawy joined Monaco on loan at the start of the season from Italian outfit AC Milan but has failed to score in 15 Ligue 1 appearances this term.
Vasilyev said the principality club are in negotiations over El Shaarawy's exit amid interest from Roma.
"El Shaarawy is not any more in the project of the head coach of Monaco," he told the club's official website.
"We are currently negotiating to find a solution. Maybe it’s not a comeback in Milan. It could be elsewhere.
"We are not planning to make him play again. That’s not planned. He’s a great player with a lot of qualities. He’s an international player with Italy. We need to find the right solution for everyone. I hope we will."
The forward will be hoping for a quick solution to the situation with Euro 2016 quickly approaching.
"We are discussing with Milan, his agent and other clubs in order to find something," Vasilyev added.
"I would like for Stephan that he keeps playing in order to be at the Euro because I know that the head coach, [Antonio] Conte, is thinking of him.
"I would like to find the right solution for him."
Milan CEO Adriano Galliani confirmed the negotiations with fellow Serie A club Roma.
"There are negotiations with Roma for El Shaarawy," Galliani told Mediaset Premium.
"If we were to reach an agreement with Roma, we’d have to sit around a table to find a solution."