The Egyptian winger is reportedly unhappy with his club for not doing more to allow him a move to Manchester City in the January transfer window.
And Poyet is of the belief that if a great offer comes in for a player, a club should be able to help them move to a team with a bigger profile or ambition.
"It's very tough situation," Poyet said to talkSPORT.
"I think Mahrez gave Leicester something that a few years ago no one, and I mean no one, thought would be possible, to win the Premier league.
"He has been a great servant of the club and has had a unique opportunity to go to a team that is playing incredible football [Man City].
"The player wants to go, logically, the transfer fee is more than good, but for reason somebody has said 'no'.
"I think it is clear the person who has said 'no' has to take responsibility and should not put that on the player.
"As a footballer, when the opportunity comes to go to a club you would love to join, and it doesn't happen, it's hard to take.
"What I would say to Leicester fans is, please, understand the player as well, don't only think about one side, the club.
"It's very difficult and, if that happened to me, of course I would be very, very upset."