Exeter have enjoyed a meteoric rise in English rugby, climbing from the lower leagues to win the Aviva Premiership.
The Chiefs beat Wasps at Twickenham in May and they will return to action when the season restarts in September.
Steenson will be expected to line up against Gloucester during the Friday night game at Kingsholm and he admits that the new challenge excites him.
He told reporters: "It's a completely brand new experience for us.
"We are probably not everyone's second favorite team any more, we are not the plucky underdog, which is a completely new thing.
"It's nice to come in and have a new challenge. The club has always been about progressing and there is so much more the club can achieve and do.
"We haven't performed that well in Europe, which is something we will definitely talk about."