Opposition parties have pushed for the vote to be held in secret, hoping to encourage ANC lawmakers to vote Zuma out of office after a series of corruption scandals.
But the president retains strong support from many lawmakers in the African National Congress, which led the fight against apartheid.
In the last two years, Zuma has easily survived three votes of no confidence and a separate parliament vote to remove him from office.
Mbete, a Zuma ally, will decide whether the vote is secret after a case brought to the Constitutional Court by opposition parties, who have called for ANC lawmakers to "vote with their conscience".
More From This Section
The no-confidence vote was initially scheduled for April but delayed to allow the court to rule.
The ANC has vowed to defeat the motion.
Zuma is due to step down as ANC head in December and as national president ahead of the 2019 election.