A member of the Dáil committee that could be asked to carry out the banking inquiry has called for it to be delayed to avoid prejudicing any criminal trials next year.
Independent TD Stephen Donnelly said Anglo must be within the terms of reference.
He said the bank constitutes about half of the money that the Irish people have paid out.
"Any Oireachtas inquiry that doesn’t have Anglo Irish Bank within the terms of reference is a complete waste of time," he added.
Independent TD Shane Ross has said it is "frankly ridiculous" to proceed with the inquiry without considering Anglo Irish Bank.
He said the bank's footprints were all over the period leading up to the bank guarantee.
But Taoiseach Enda Kenny has insisted the court proceedings against former Anglo Irish Bank executives and the proposed banking investigation are distinct and separate.
The inquiry is to examine events surrounding the bank guarantee, the role of banks and auditors, and the role of State institutions.
The Dáil took the first steps towards setting up the inquiry today by passing the order that will establish the investigation.
Fianna Fáil Finance Spokesman Michael McGrath has said the Government should examine practices at Anglo Irish Bank as part of any future banking inquiry.
Mr McGrath said it was possible to conduct a statutory banking probe without affecting criminal cases currently before the courts.
Sinn Féin Finance Spokesperson Pearse Doherty has welcomed the planned inquiry.
He said it was up to the members of the Oireachtas to decide which structures and terms of reference should be put in place.
He said an effective banking inquiry could not go ahead without scrutiny of Anglo Irish bank.