The Germany international impressed under Pep Guardiola last season, helping the Citizens win a Premier League and EFL Cup double.
Mahrez, meanwhile, was instrumental as Leicester City won the 2015-16 title against all odds and will be eager to cement a place on the Man City wing next season.
"I'm totally happy for it - I like competition," Sane told The Mirror.
"He's a really good person. When I first met him here, he's really nice, open, and a really class player.
"I think everyone's seen it before in Leicester that's why I'm really happy that he's here."
Sane missed out on World Cup selection with Germany despite picking up the PFA Young Player of the Year award.