Home Affairs portfolio committee chairman Lemias Mashile told MPs that Apleni was briefing another committee in Parliament on Tuesday morning, triggering some tense exchanges.
Democratic Alliance MP Haniff Hoosen said an explanation on how the Indian-born family was granted early naturalisation was required not only from the DG but from former Home Affairs minister Malusi Gigaba.
Gigaba, said Hoosen, needed to prove that he acted "impartially and fairly and within the law".
Economic Freedom Fighers MP Hlengiwe Hlope was perturbed by quotes she attributed to Mashile in which he reportedly said that he was not going to allow a request from the DA that Gigaba be called before the committee.
"It is not right for you to defend everything," said Hlope.
The ANC's Maesela Kekana said the committee should not defend Gigaba and Apleni but should summon them to appear before MPs.
"For sure as all of us as members of this committee, we are not here to compromise our integrity. We are here because of people and this thing of the DG and the minister that they are not available is total disrespect of this committee. We are not here to defend anybody," he said.
"I propose let us make sure that we summon the minister and the previous minister and the DG to come. It's not something they must volunteer to come here. They must know they account here ...let them get the summons."
Mashile said the reason the DG had been called first was to explain the "hard facts" around how the Gupta family, who have close ties with President Jacob Zuma, was granted early naturalisation.
The minister could then be called.
"I'm saying we expect the DG to tell us the hard facts of the matter, to confirm the hard facts of the letter we have seen in the media, is it correct, is it authentic?"
MPs later agreed, some reluctantly, to first deal with other issues on the agenda while a message is sent to Apleni to appear before them later in the day.