Former Anglo Irish Bank chairman Sean FitzPatrick and former chief executive David Drumm have been directed to attend the Banking Inquiry on 29 July.
Both were notified three weeks ago they would be called. It is understood they have each acknowledged the correspondence from the inquiry, although they have not confirmed their attendance.
The inquiry has powers of compellability, although this would not extend outside the jurisdiction.
Irish Nationwide's Chairman Michael Walsh has been directed to attend on 2 September. It is understood he has also responded to the committee.
Other bankers from Irish Life and Permanent and EBS have also been given dates to attend.
Earlier, former financial regulator Liam O'Reilly told the inquiry that the system of regulation did not work and more robust and intrusive measures should have been taken.
He said he deeply regretted the failings in the system were not recognised during his time.
Mr O'Reilly joined the Central Bank in 1967 and was financial regulator from 2002 to 2006.
He said there was no belief in the Financial Regulator that conditions might deteriorate to such a level that the banking system would succumb to a traumatic shock of the magnitude that occurred in 2008.
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Mr O'Reilly said the crisis was an iceberg he did not see when he left the Regulator in 2006, saying he thought everything was okay.
He agreed that in hindsight, the crisis was embedded by then but he did not have a sense of it at the time.
It never occurred to him to apply mandatory lending limits on banks.
He said they were looking at ways of trying to dampen credit, but there was confusion about concentration limits and these guidelines had fallen into abeyance and were not applied.
They were too patient in dealing with one bank in breach of regulations and he regretted that he did not just place a condition on it.
Mr O'Reilly said sanctions should have been applied quicker under his successor Patrick Neary.
He said there seemed to be a "long fingering" of the implementation of sanctions.
Fianna Fáil's Michael McGrath challenged Mr O'Reilly on blaming the system, saying people find that frustrating.
He asked if it was adequate to blame the system.
Mr O'Reilly said the system was there, they operated it and he didn't have a sense of danger.
They had difficulties retaining staff in a competitive environment, Con Horan was a good staff member and he valued him. He was not slow in giving his opinions.
Mr O'Reilly said that in 2004 he was "cynical about whether they [the banks] were applying the policies that they said they were applying."
He said moral suasion was all that was applied but the regulator should have done more.
He said he was worried about loan deposit ratios in the period from 2002 to 2008 and the question all the time was whether in a market economy "you can put in quantitative limits."
Mr O'Reilly said the question of exposure was far from the mind during this period because "there didn't seem to be a consciousness at all that credit markets could freeze up and all of sudden you're in trouble."
Mr O Reilly said he became a member of the board of Merill Lynch Ireland in 2007, but the specialist team that subsequently advised the government came from abroad.
In September 2008, he said he joined the board of Permanent TSB.
Mr O'Reilly said he was very careful as to what institutions he got involved in and made sure there were no issues around the time that he was a regulator.
He said he rarely met with the CEOs of banks between 2002 and 2006, and in total maybe five or six times.
Asked if the banks were regulating themselves, he said there was a need for much more intrusive regulation.
Asked if the regulation was naive, he replied "And a failure."
Mr O'Reilly said "contrarians" within the banks were listened to but they weren't in the majority.
He said the majority dependence the regulator had, was with the internal audit departments and the external auditor for checking out what banks were doing.
He said compliance problems didn't occur generally as a result of misbehaviour by senior executives.
He said banks are very proud of their own risk management systems and they do not like to be criticised and when the regulator find fault they sometimes push back but then "we push back."
He said that was the principles based approach but now the system was "I don't care what you say, do it."
Fine Gael's John Paul Phelan asked if Mr O'Reilly had a conflict of interest in taking up board positions on Merrill Lynch and Permanent TSB after his retirement.
Mr O'Reilly said he never ever worked to make a lot of money and he was a public servant.
He could understand the perception but he had a constitutional right to work.
Socialist Party TD Joe Higgins asked if they were gentlemen regulators dealing with street brawlers?
