It follows days of criticism after Mr Hogan apologised for his attendance at the Oireachtas Golf Society dinner on Wednesday night.
Gardaí are investigating the gala dinner at a hotel in Clifden, Co Galway for alleged breaches of public health legislation.
In a statement issued tonight, Micheál Martin and Leo Varadkar say Mr Hogan's apology "came late", and he still needs to give a full account of his actions.
“The Taoiseach and the Tánaiste did speak with the Commissioner today and asked him to consider his position,” a Government spokesperson said in a statement.
“They both believe that the event should never have been held, that the Commissioner’s apology came late and that he still needs to give a full account and explanations of his action.”
Mr Martin and Mr Varadkar have not spoken to European Commission president Ursula von der Leyen, who would have the power to remove Mr Hogan from office.
Mr Hogan’s appointment as Commissioner was ratified by the European Parliament after being nominated by the previous government.
Sinn Féin's health spokesman David Cullinane says the scandal has ruined the Government as a voice of authority.
