Ronny Deila will leave the Scottish Premiership side at the end of the season.
The Bhoys are set to appoint a new manager ahead of next season with Ronny Deila set to step down at the conclusion of this campaign.
And Lawwell explained that he wanted the Norwegian's replacement to be able to help Celtic make an impression on the continental stage.
"We want somebody to create a team that will participate in the Champions League. We're a Champions League club in everything we do," he told The Scottish Sun.
"We're a top-class professional football club and that being the case we should be in there."
Though Deila has won the Scottish Premiership title during both of his seasons at Parkhead, he has been unable to inspire the side to make an impact in Europe. They have failed to reach the Champions League group stage in either year, and subsequently failed to win a single game before their Europa League exit this season.
With the Bhoys dominating the domestic league, it is arguably Champions League success that will enable the club to keep hold of their strongest players as it provides the challenge that several of those squad members crave and may have previously felt that they needed to move elsewhere to experience it.