Grosjean was eliminated by Sweden's Jonas Bjorkman, 7-6 (7-2), 1-6, 6-4, while Clement squandered a pair of match points en route to a 7-5, 3-6, 7-6 (8-6) defeat at the hands of Belgium's Xavier Malisse.
Grosjean had a 5-3 lead in the opening set before losing the tiebreaker, then dropped the decisive set after rallying from a 0-4 deficit.
"This is the first time I ever won a match here in Lyon," Bjorkman said. "This has definitely been a huge confidence boost for me. I know when I play well that I can be in the top 20. I had high hopes for this season, but mentally I have been struggling with the high expectations I put on myself."
Clement sent a backhand volley into the net on his first match point and Malisse erased the other by firing an overhead well out of Clement's reach.
Malisse will face Bjorkman in the quarterfinals in a battle of players seeking their first title of the season.
In other second-round matches, second seed Juan Carlos Ferrero rolled to a 6-2, 6-2 victory over Germany's Nicolas Kiefer and third seed Marat Safin of Russia swept past Israeli qualifier Noam Okun, 6-3, 6-4.
Safin, trying to climb from 11th place in the ATP Champions race, faces Max Mirnyi in the quarterfinals after the native of Belarus ousted Hicham Arazi of Morocco, 7-6 (7-4), 6-2.
Awaiting Ferrero is Morocco's Younes El Aynaoui.
Ivan Ljubicic of Croatia earned the final quarterfinal spot with a 7-5, 7-5 triumph over France's Julien Benneteau.