Algerian Mahrez has neither trained nor played with the Leicester team since transfer deadline day, when he failed to secure a move to the Premier League leaders.
"I hope Riyad can get his head right and come back with us. The best way is to come back and enjoy his football," Puel told a news conference on Thursday.
"Riyad is a magnificent player and he loves football. He loves his relationship with his friends and team mates. He needs to come back and to play, to touch the ball."
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Puel would not comment on whether Mahrez, who has missed three games and five training sessions, has been granted permission to be absent by the club.
"I don't want to comment and give details of our relationship with Riyad. It is important to all details remain inside the club and not in the public," he said.
British media reported friends of the player saying he has been "depressed" since the deal failed to go through.
Asked whether he could understand the frustration felt by Leicester City fans at Mahrez's absence, Puel said: "I can understand all this speculation. I think it is important in this difficult time to keep all players, and club, united. Together, to look forward and continue the hard work on the pitch."
Puel's comments came as British media reported that Manchester City have cooled their interest in Mahrez and do not intend to bid for him again.
Former Leicester midfielder Dennis Wise criticised Mahrez on Thursday.
"He has let himself down, his team mates down. He needs to get back playing football again," Wise said on Sky TV.
Shinji Okazaki and Wes Morgan will also miss Saturday's game.
"Okazaki will not be available. He received a kick on his knee and won’t be available for two weeks," Puel said.
(Reporting by Neil Robinson; Editing by Ed Osmond)
