Rupert Murdoch and Rebekah Brooks have been invited to appear before the culture, media and sport select committeeCredit: Barry Batchelor/PA
However, the committee has issued a summons for News Corporation boss Rupert Murdoch and his son, and chairman of News International, James, who both claimed they were unable to attend.
The committee added that James Murdoch did offer to appear at an alternative date, on 10 August at the earliest.
Two days ago the committee requested the trio appear before it on Tuesday next week and this morning received responses from all three.
While Brooks accepted the invitation, Rupert Murdoch indicated he is "unable to attend to give evidence" and James Murdoch said he is unable to appear on the specified date.
But the committee said it has "made clear its view that all three should appear to account for the behaviour of News International and for previous statements made to the committee in Parliament, now acknowledged to be false".
As a result the committee has decided to summons Rupert Murdoch and James Murdoch to appear before it on Tuesday at 2.30pm.
In his letter to the committee, Rupert Murdoch said he was "fully prepared" to give evidence to the judge-led public inquiry announced by the prime minister, saying he will be "taking steps to notify those conducting the inquiry of my willingness to do so".
"Having done this, I would be happy to discuss with you how best to give evidence to your committee," the letter adds.
In her letter accepting the request, Brooks indicates there may be some details she will not be able to discuss due to the ongoing investigation.
"I understand that various select committees have approached the police over time in relation to this and other cases. The police's position has been to co-operate where this did not directly impact on the investigation in question.
"In those cases where it did potentially impact, the police have historically declined to comment at that stage. Our understanding is that this approach has not been challenged.
"Given that we are in the midst of an investigation, and we do not want to prejudice it, I hope you will understand why we feel it would not be appropriate to respond to such questions at present in order to be consistent with [the] police's approach, and that as a result this may prevent me from discussing these matters in detail."