Fine Gael's Paschal Donohoe has said that there is "a chance" a government may be formed before Christmas, as he praised the impact of his party leader Simon Harris.
Mr Donohoe held on to his seat in the Dublin Central constituency, and told his reporters following his win: "Overall the centre has held up in Irish politics. It has held up."
"I'll be going in to make the case for Simon Harris as Taoiseach," he said.
"I believe he has done extraordinary work during the campaign and in particular in the final days of the campaign he made the decisive difference to our support."
Asked if there would be a government formed before Christmas he said: "Well, yes, there is a chance, but we do have a lot of work to do."
Earlier, the former deputy leader of the party said it will be a "very difficult proposition" for Fine Gael to re-enter a coalition with Fianna Fáil without a rotating taoiseach.
Fianna Fáil is likely to increase its seat lead over Fine Gael compared with the 2020 election, which saw the parties enter a coalition on the basis that the holder of the taoiseach position would be exchanged midway through the term.
Mr Coveney, who retired from politics at the end of the outgoing Dáil term, said: "There are a number of ways that you can define numbers.
"There are the number of seats in the Dáil, but there's also the percentage vote that the parties have got nationally, and the percentage vote that the parties have got nationally is more or less the same - that's an important mandate that certainly Fine Gael will carry into any discussions.
"I think it would be a very difficult proposition for Fine Gael to move into a coalition government, having just come through a coalition government where a rotating taoiseach has worked very successfully and has contributed to stability in government.
Speaking to reporters at a Cork count centre, he said it will be a matter for the party leaders and for the parliamentary parties concerned, but from a Fine Gael perspective, "it would be very difficult for them to consider going into government without having a taoiseach for part of that government".
Coalition with smaller party 'more stable'
A coalition with a smaller party would be "more stable" than an agreement with independents, Mr Coveney added.
He said Fine Gael and Fianna Fáil had proven they are "compatible partners in government" but added that he suspects a third partner would be needed.
He said he was speaking personally as he was not a "decision maker any more", but added: "I think a party is generally more stable than a group of individuals, although I have been involved in coalition governments with both.
"And I have to say, the coalition government we had with a group of independents worked pretty well - and there are quite a number of independents who are going to get elected who, if you like, are from a sort of Fine Gael-Fianna Fail gene pool."
Attention has turned to whether Labour or the Social Democrats could become a junior partner in a coalition.
Mr Coveney, who was minister for foreign affairs during Brexit negotiations between the UK and the EU, said he thinks both the Social Democrats and Labour will come back with a reasonably strong performance in this election.
"I hope both of those parties will want to be in government, and we'll talk with an open mind and with realism in terms of how you can put a government together with Fine Gael and Fianna Fail."
More generally, Mr Coveney said the results so far had been an "awful lot better" than previously predicted.
He said party leader Simon Harris "can take credit for that", but said the Taoiseach had a difficult campaign.
"It's been a very tough campaign for Simon Harris. Things have gone wrong that he probably couldn't have anticipated and didn't expect.
"I think he's shown remarkable resilience and toughness, quite frankly, which is what you need as a political leader these days.
"To actually find a way through that and still ensure that Fine Gael got a good result, growing our representation of the Dáil and bringing in a whole new crop of new TDs."
Mr Coveney conceded that Fine Gael had not done as well as Fianna Fáil in seat gains, but said his party had done well considering challenges around retiring incumbents.
"We'll be at in and around 38 seats, I hope. Fianna Fáil might be close to 48 seats," he added.
