Pep Guardiola's side reportedly made a £65 million bid for the winger as they looked to add depth to their attack, following an injury to Germany international Leroy Sane.
However, Leicester purportedly demanded over £95m to part with their winger, who was integral to the Foxes' Premier League title win in 2015-16.
Mahrez has been in inspired form this season, scoring eight goals and providing eight assists in 24 appearances, but handed in a transfer request in an attempt to force the move through.
The Foxes remained unmoved, however, and Madjer believes that his star player should be playing at a higher level.
"Mahrez currently deserves better than Leicester," he told Goal. "But we must not forget that it was Leicester who put him on the right track.
"I think a serious conversation with his board is needed. He wants to go to a bigger club in the English league. I do not know what really happened about his transfer."
It has been reported that Mahrez has missed training four times since the breakdown of his transfer, but Madjer has advised him to return to work, as it may lead to a move in the summer.
"Me, I advise him to resume training and work hard, even if the transfer has not been resolved," he said.
"And we must not forget that there are the games of the national team (that will come soon) and a preparation. May he come back to us in great shape. I hope that later the doors will be opened by a bigger club."
Claude Puel's side are eighth in the Premier League, though they are closer to the bottom three than the top four with 12 games remaining.
Mahrez could play against his prospective employers in the next round of fixtures, with Leicester set to travel to the Etihad Stadium on Saturday.