The Hoops won the Ladbrokes Premiership for the seventh successive season while their rivals ended the campaign third under caretaker boss Jimmy Nicholl, who makes way for the former England and Liverpool captain.
Rodgers, who managed Gerrard at Anfield, was speaking at Lennoxtown after he picked up the Ladbrokes Premiership manager of the year award.
Asked if he expected the arrival of Gerrard at Rangers would make it even tougher next season, the Northern Irishman said: “I would expect whoever was in charge at Rangers for it to be tougher.
“That’s how football works, each year it gets tougher and more difficult and that will be no different, whoever was taking charge, whether it was Steven or anyone else.
“The competition was always there and it will be next season. We have to improve again next season which we will do, both individually and as a team.”