Fianna Fáil finance spoke-sman Michael McGrath said the move should take place to shed light on what happened in the lead-up to the economic crisis.
Speaking after Tuesday night’s ruling in the US, Mr McGrath said it was “vital” that Mr Drumm came before the group.
While accepting a potential extradition could mean the banker cannot attend, Mr McGrath said that the request must still be made, as Mr Drumm was “uniquely positioned” to reveal what really happened in 2008.
However, banking inquiry chairman and Labour TD Ciaran Lynch said the move would be premature.
He said that the inquiry must firstly obtain all details before deciding who should appear.
The inquiry has compelled all seven banks covered by the guarantee to provide by February 3 a raft of details, including any links between senior officials’ bonuses and the volume of property loans for property.
Failure to do so could result in legal action being taken.
The banks involved had until today to file objections to the request, but none were lodged by yesterday evening.
Meanwhile, Fianna Fáil senator Averil Power has written to US secretary of state John Kerry and US ambassador to Ireland Kevin O’Malley asking for Mr Drumm’s US visa to be revoked.
Ms Power called on Taoiseach Enda Kenny and Foreign Affairs Minister Charlie Flanagan to make a similar call, saying that the US would not allow disgraced financier Bernie Madoff to remain in Ireland if the situation was reversed.
However, given the fact Mr Drumm is in the US on an investor visa, it is highly likely any attempt by him to extend his stay will fail — making such a request unnecessary.
State seeks to extradite ex-bank chief
The State has launched an attempt to extradite the former chief executive of Anglo Irish Bank after he failed to be declared bankrupt in the US.
David Drumm moved to Massachusetts in 2009, six months after resigning as chief executive of the now defunct bank.
Along with two others, he stood down after hundreds of millions in directors’ loans were uncovered.
The 48-year-old has since refused to return to Ireland to be questioned about the events leading to the collapse of the bank, which was later nationalised before being wound up.
Drumm fought a four-year legal battle in Boston to write off his own debts of €10.5m.
But a US bankruptcy court rejected his claims in no uncertain terms, ruling that the ex-banker was “not remotely credible”.
During the proceedings, the Irish Bank Resolution Corporation — formerly Anglo Irish Bank — fought his claims for bankruptcy.
The institution, which is being liquidated, is owed €9m by Drumm.
It claimed he had knowingly and fraudulently put assets beyond the reach of his creditors, mostly by transferring them to his wife.
The judgment in the US leaves him facing financial ruin.
Separately, the DPP has recommended a number of charges against Drumm after a years-long probe into Anglo Irish Bank by the Garda Bureau of Fraud Investigation and the Director of Corporate Enforcement.
It is understood that a formal extradition request has been sent through diplomatic channels from Dublin to Washington.
Senator Averil Power said she has written to the US secretary of state John Kerry urging that Drumm’s visa be revoked immediately.