Congress of the People Parliamentary leader Mvume Dandala on Thursday gave notice that he would move a vote of no confidence in President Jacob Zuma.
Speaking in the National Assembly, Dandala said this was because of Zuma's "failure to live up to the expectations of a broad spectrum of South Africans".
The motion will be noted on the order paper, but there is no indication of when, if at all, it will be debated in the House.
In a statement issued earlier, African National Congress (ANC) Chief Whip Mathole Motshekga said the motion was "nothing other than a cheap shot at publicity, aimed at salvaging its [Cope's] sinking public profile".
"After ten months in Parliament, the party has nothing to show to the electorate for its presence in this institution.
"Charades such as these may provide the party with the 15 minutes of fame in the media, but they regrettably make a mockery of our democratic system and the institution of Parliament," he said.
Zuma had "demonstrated exemplary leadership by apologising to the nation and therefore there is no basis, constitutionally or otherwise, for such a motion".