The Fine Gael leader said “anybody” who has been asked to attend by the high-profile inquiry “should co-operate fully and completely”, adding that “in my personal view that includes Mr Drumm”.
The exiled banker told the inquiry on Wednesday night he will not attend in person this week.
This is because of outstanding Garda interest in the activities of the former banker, who went to Boston, Massachusetts, after the bank guarantee.
Mr Drumm, who has previously been the subject of mooted extradition requests to return to face questioning in this country and tried to file for bankruptcy in the US, offered to give his evidence by video link in correspondence on Wednesday night.
However, while the issue is still being considered by the investigation and will not be ruled on until legal advice from the DPP and its own experts is given on Monday night, a number of members will refuse to take part.
In a statement, Fianna Fáil finance spokesperson Michael McGrath said the idea goes against everything he stands for and would mean a central player could give evidence “on his terms” while refusing “to co-operate with the Irish criminal justice system”.
Fine Gael TD Eoghan Murphy and party colleague, Senator Michael D’Arcy, also confirmed they will refuse to take part.
While Fianna Fáil senator Marc MacSharry and Sinn Féin finance spokesperson Pearse Doherty are both waiting until DPP and inquiry legal advice is finalised on Monday night before making a decision, the politicians were last night understood to be moving towards the same position.
The situation means at least five of the 11-strong inquiry team are opposed to varying degrees to the Drumm video link request, causing serious questions over how the investigation can grill the banker as he is highly unlikely to return to Ireland willingly.
While the inquiry will be given its own legal experts’ final view of the matter in correspondence on Monday night, it is understood two in-house senior counsel who spoke with it during a private meeting on Thursday evening gave contradictory views.
One is believed to have told the investigation to “press ahead” with the statement and video link, while another suggested using either may impact negatively on future legal matters involving Mr Drumm in Ireland. A number of inquiry members are believed to feel this is the only way to ensure a key player in some way faces up to his responsibilities.
The inquiry has been beset with difficulties in gaining access to senior Anglo officials, due in part to an ongoing court case.
The inquiry previously met with ex-ECB president Jean Claude Trichet on neutral territory after separate claims about attending at Leinster House, a move some inquiry members believe was worthwhile and others feel undermined their work.