Montreal is proposing a five-year contract with an average annual value of $8.454 million for the 21-year-old center.
Aho has signed the offer sheet, Canadiens general manager Marc Bergevin confirmed. Carolina now has seven days to match.
''He wants to come to Montreal," Bergevin said. "He sees our youngsters coming up in the organization and he wants to be a part of that. We're proud, but there's still a waiting period.''
If the Hurricanes don't match, they'll be compensated with a first-, a second-, and a third-round pick.
Signing bonuses would reportedly make up most of the deal.
Carolina general manager Don Waddell said he was "actually surprised" the value of the offer sheet wasn't higher, according to Hurricanes team writer Michael Smith.
Waddell added there's no reason to match immediately because it would help the Canadiens sign other players, according to NHL.com's Tom Gulitti.
Aho posted 30 goals and 83 points in his third season with the Hurricanes.