The Tigers won their 10th title courtesy of a 37-17 victory over Northampton in the Premiership final at Twickenham in May, and they will be eager to win more silverware come the business end of the forthcoming season.
However, Cockerill expects a tough campaign, starting with their Premiership opener against Worcester Warriors on September 8 at Welford Road.
The Warriors finished second from bottom last season with 33 points but Cockerill expects Dean Ryan's side to give the Tigers a stern test, and he expects plenty more to come over the course of the season.
"You see what's happening with Worcester; Dean Ryan will have a well drilled Worcester team," he told the Aviva Premiership website.
"We'll play them first game and they'll come here believing they can win, and with a new coach, a new set up and new players, we're going to have to be right on it.
"[John] Wells will have a very well drilled squad at Newcastle Falcons, and London Irish have changed what they're doing a little bit and have got a very big squad that won't lose many players.
"Every year if you look at Gloucester and those sides and the sides that have been the traditional top four over the last couple of years I think it gets tougher and tougher and that's what it should be, that's the attraction of the Premiership."