The Frenchman has just one year left to run on his current contract and he admits there is a lot of interest in him after an impressive season.
"The negotiations have lasted a little. We will have a new meeting on May 29," Evra said in L'Equipe.
"Now it is up to the club, who know what we have already agreed. We will see.
"A lot of clubs are attracted to me. It is quite logical as I did a lot on the pitch.
"Anyway, I hope we will find a solution as soon as possible."
The left-back has been included in Raymond Domenech's initial squad and is looking to feature heavily in Austria and Switzerland.
He said: "Euro can be a very important step in my career. It will be my first tournament with the French team.
"I am confident, and at the group's disposal. I hope I can finish the season as European champion with Les Bleus."