The motion of no confidence debate against President Jacob Zuma was scheduled to be heard on April 18.
"The EFF holds the view that the outcome of the Constitutional Court has a direct bearing on the possible nature of the motion of no confidence. We therefor make a rational request that the Speaker the sitting on April 18 to a date that would be decided after the Constitutional Court hearing and decision," EFF chief whip Floyd Shivambu said in the request to Mbete.
The UDM has also asked Mbete to postpone the debate.
The party approached the highest Court in the land to convince that next week's vote of no confidence Zuma should be a secret vote by MPs.
The African National Congress (ANC) has indicated none of its MPs would veer away from the party line and "vote with a conscious together with opposition parties".
Opposition parties support the Democratic Alliance (DA) tabling of motion of no confidence against Zuma, in another attempt to unseat the president.
Various attempts by the DA to push through the motion against Zuma failed as the ANC used its majority in the National Assembly to block such a move.
Shivambu said the UDM's Constitutional Court application was important for the motion, hence the request for a postponement.