The contract is for one year, close to $1 million, the report says.
Franson, 25, played in 57 games for the Maple Leafs last season, with five goals and 16 assists. He was a minus-1 after playing the previous two seasons for the Nashville Predators.
Comcast SportsNet New England reported Saturday that defenseman Mike Mottau also is expected to sign a one-year contract with the Maple Leafs.
Mottau, 34, had been playing for the San Antonio Rampage, the Florida Panthers' affiliate in the American Hockey League.
He has played 313 NHL games after making his debut in 2000-01.