The current champions have been placed as the strong favourites to make it three titles in a row this season, but Forrest admits that it adds "more pressure on to the players".
"I think it maybe adds more pressure on to the players. But we have handled that in the past and I'm sure we can handle that again," he told PA.
"A lot of SPFL teams have good squads this season and just look at St Johnstone doing well in the Europa League on Thursday (1-0 win away to FC Minsk), so I think it will be harder this season.
"We have signed a few new players and the squad is looking really strong just now. We still have a few players to come back from injury and once we do, we will be looking good."
Celtic open their campaign against Ross County on Saturday.