The Leicester Mercury newspaper said the Foxes turned down City's final bid of 65 million pounds plus an unnamed player for Mahrez, a deal that would have broken the Manchester club's transfer record.
Leicester valued Mahrez at around 95 million pounds, but City were unwilling to meet that fee, according to British media reports.
Mahrez was reportedly keen to join Pep Guardiola's side, with British media reporting that he had been left depressed by his failure to secure the move.
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Puel said he hopes he can reintegrate the 26-year-old Algerian back into his squad. Mahrez submitted a transfer request, missed training and was not included in Wednesday night's 2-1 defeat at Everton.
"I was ready to accept the situation. Move or no move, it was a deal between the clubs," said Puel.
"It was a difficult situation," added the Frenchman.
"(His staying is) a good thing for the club. I hope he can come back with a smile, and see him in the training session to prepare."
City manager Guardiola confirmed that the deal would not go ahead.
"Mahrez is a Leicester player. I don’t talk about it too much. The transfer window is over so the guys who are here will finish the season," he said.
This is the second disappointment for Mahrez on transfer deadline day. In August he was widely reported to be at an airport ready to fly to Europe if any clubs came in for him.
(Reporting by Neil Robinson, editing by Ed Osmond and Toby Davis)
