Cleverley, who started the season in fine form but has struggled with injury since then, is likely to feature at the Olympics if he is not included in the full England squad for Euro 2012.
Anyone who is on the plane to Poland and Ukraine this summer will not be considered for duty at the Games.
Ryan Giggs and Beckham are among 184 players who have confirmed to the Football Association they are willing to play for Team GB, and Cleverley hopes to line up alongside the United legends.
"It is something that everyone strives for, to experience new things throughout your career, so the Olympics would be different," Cleverley told the Mirror.
"I always looked up to Beckham when I was young. To play alongside him would be great."