Mr Cowen has faced recent scrutiny about a meeting with the head of Anglo Irish Bank shortly before he announced a multi-billion euro bank guarantee. Several Fianna Fail backbenchers have called for Mr Cowen to resign.
On Saturday, one Fianna Fail member of parliament called for the party to unite behind the foreign minister, Micheal Martin. Noel O'Flynn said he had been contacted by the prime minister on Saturday afternoon and had told him that his leadership had not worked.
Former Anglo Irish chairman Sean Fitpatrick, who Mr Cowen met at a golf course in 2008, is still subject to official scrutiny over his role in the collapse of the bank and has previously been questioned by police.
The Taoiseach (prime minister) has denied that he discussed bank matters with Mr Fitzpatrick at the golf course and insisted that he had not done political favours for any financial institution.
On Thursday, Mr Cowen made clear that he did not intend to resign and said the process of talking to his parliamentary party should take about 48 hours.
He is also facing a motion of no confidence, tabled by the opposition Labour Party, in parliament next week.
Mr Cowen, who took power in 2008, has been under pressure since Ireland was forced to seek an international financial bail-out late last year.
Following the intervention by the EU and the IMF, there were rumours of an internal push against him but his critics pulled back at the last moment.
His coalition government has only a slim minority in parliament and a general election is expected in the spring.
