MPs were engaged in a tussle with the head of MI5 over whether the spy chief would give evidence to parliament in public to a Commons select committee.

Andrew Parker has agreed in principle to appear before the home affairs select committee, but neither the status of the hearing, whether it would be in public or private, nor its timing had been confirmed. The chairman of the committee, Keith Vaz, wrote to Parker inviting him to give evidence on 17 December, and asking him to confirm his attendance by next Monday.

The suggestion of a date came after MI5 made an official response to Vaz's announcement that the director general of the security services had been summoned to appear before the committee. MI5 made clear that Parker was not going to be rushed into agreeing the terms of his appearance, choosing to emphasise a point of procedure: "The director of MI5 has not received a request from the home affairs select committee to attend a hearing," said a Whitehall source. "When he receives one, he will consider it. But at the moment, he hasn't received an invitation. He has not had sight of any request to appear next week or next year."

Parker's public appearance before the committee would be an unprecedented step in the history of the oversight of the intelligence agencies. The appearance of the chiefs of the three intelligence services, MI5, MI6 and GCHQ, before the intelligence and security committee (ISC) last month, was itself the first time such figures had given evidence in front of the cameras. But the members of the ISC are appointed by the prime minister and not by parliament, and there was criticism when it emerged after the hearing that the questions asked had been prearranged with the security chiefs.

In his formal invitation, Vaz says that the committee "is mindful of the need to avoid encroaching unnecessarily on to territory which properly belongs to the intelligence and security committee", then says: "But your evidence will make an important contribution to the committee's examination of the counter-terrorism policies of the Home Office, their budgets and administration."

The tussle took place as it emerged that the "big five" American internet companies say greater transparency is needed over UK official requests for access to their users' data to "maintain confidence that powers are not being abused".

Google, Facebook, Yahoo!, Twitter and Microsoft told the committee's inquiry into counter-terrorism that the existing legal framework on access to communications data or metadata could be reformed "without the need for wholly new legislation" – believed to be a reference to the so-called "snooper's charter".

In written evidence, the web companies said that transparency is important, as is encouraging a full public debate "as public concern grows around the world about the scale of digital surveillance". The evidence emerged a day after Alan Rusbridger, the editor-in-chief of the Guardian, defended before the committee the publication of disclosures based on files leaked by Edward Snowden that have revealed the "staggering" scale of Britain and America's secret digital surveillance programmes.

Cressida Dick, Scotland Yard's assistant commissioner in charge of counter-terrorism, told journalists that she agreed with the head of MI6, Sir John Sawers, that terrorists would be "rubbing their hands with glee" at the disclosures. "If you look at some of the things which have been apparently revealed, you can see immediately how that would affect people's way of working. It will be putting, I believe, people's lives at risk, whether they are agents, sources or members of the public," she said.