This follows a raft of complaints against Mkhwebane over the past few months, with calls that she be removed from office.
Yesterday, Mkhwebane denied snubbing the meeting of the portfolio committee on justice, saying she had a family emergency to attend to.
Since Mkhwebane took office in October 2016, she has been taken to court several times and MPs have questioned some of her findings.
In a robust meeting of the portfolio committee on justice, MPs said they wanted to determine if the inquiry was necessary and, if so, decide on the scope and witnesses.
MPs were also left fuming after Mkhwebane missed yesterday’s meeting. They said the fact that no presentation was sent to them in advance, which is the practice in the House, showed she had no intention of coming to Parliament.
The committee's chairperson, Mathole Motshekga, said they would decide next week whether the inquiry will go ahead, because of scathing judgments against Mkhwebane as well as complaints from the opposition, a law firm and other parties.
“We have not decided to hold an inquiry, but we are asking her to appear before the committee to determine if we are proceeding with the inquiry. There are court judgments (against her); there is another letter from (a law firm); and there are observations made by ourselves.
"I want her to be here next week to respond,” Motshekga added.
Former public protector Selby Baqwa, now a judge, completed his term as the first public protector from 1995 to 2002, after he was appointed by former president Nelson Mandela.
Lawrence Mushwana’s tenure from 2002 to 2009 was also not mired in any bid to remove him from office.
Advocate Thuli Madonsela’s seven-year tenure from 2009 until 2016 was marked by landmark findings against state officials and former president Jacob Zuma, with precedent-setting Constitutional Court judgments on his Nkandla property confirming her powers.
Other parties in Parliament have called for Mkhwebane to go. The DA and African Christian Democratic Party said she should be fired for failing to do her job.