Fine Gael ministers, TDs and senators are pushing for Leo Varadkar to serve as Taoiseach first in a coalition with Fianna Fail and the Green Party.
The issue has been raised privately by a number of Fine Gael ministers in recent days, while TDs and senators also discussed it during a video conference yesterday.
It comes as Fianna Fail, Fine Gael and Green Party negotiators formally began talks on forming the next government.
Fianna Fail and Fine Gael are expected to agree a "rotating Taoiseach" deal in a new government, with Mr Varadkar and Fianna Fail leader Micheal Martin serving two-and-a-half years each.
While it is expected that Mr Martin will go first, Fine Gael figures believe Mr Varadkar should continue to lead the country through the Covid-19 crisis.
"The public might not suddenly like to see Micheal as Taoiseach and Leo gone before the Covid emergency is over," one senior Fine Gael minister said.
A second senior minister said: "There's a very compelling case for Leo Varadkar to go first. People would be surprised to wake up and find Micheal is Taoiseach."
A third minister said an incumbent should continue in office during the crisis.
Clare Fine Gael TD Joe Carey said that he would like to see Mr Varadkar continue.
"There are many people who would say to me that they would like to see Leo Varadkar being the first Taoiseach in the new administration," he said.
"We're not through this pandemic yet, and his handling of it has been very capable."
Cork North-Central TD Colm Burke said it was a matter for the talks, but added: "It's about putting the best people forward, and there's no doubt the Taoiseach has done a fantastic job. I'd love for that to continue."
Clare senator Martin Conway said: "Most if not all Fine Gael people would like to see the Taoiseach continue in the job."
Newly-elected TDs Emer Higgins and Jennifer Carroll MacNeill both spoke about the issue during a video call for TDs and senators yesterday.
Fine Gael's Seanad leader Jerry Buttimer has also expressed concerns about Mr Martin going first.
"There is general dissatisfaction in Fine Gael that Micheal will go first. Councillors are very upset about it as well," a party source said.
A Red C/Business Post poll at the weekend found that 39pc of the public want Mr Varadkar to go first, compared with 14pc for Mr Martin.
Mr Varadkar briefly referenced the poll in the Dail yesterday, while dismissing such surveys as a test of democratic legitimacy.
He previously hinted that Mr Martin will go first when he acknowledged that Fianna Fail has more seats than Fine Gael.
Separately, Mr Carey has submitted a document to the Taoiseach's office, calling for balanced regional development, including a new rail link from Shannon Airport to the existing Galway-Limerick rail line and a new light rail system to link Nenagh, Thurles, Foynes, Ennis, Gort and Athenry to the cities of Limerick and Galway.
As talks with the Greens began last night, Fianna Fail and Fine Gael both committed to reforming EU fiscal rules, which limit excessive budget deficits, in order to fund investment.
The commitment was made to the Social Democrats in an effort to lure the party into talks.