The Taoiseach Leo Varadkar says Fine Gael is ready to make an agreement with Fianna Fáil.
The two parties are working on a framework document on policy on how a new Government may operate.
If agreed, the plan will then be pitched to other parties for consideration.
The Green Party, Labour and the Social Democrats have previously said they are not interested.
The Tánaiste Simon Coveney earlier told to South East Radio that the contents of the framework may surprise some people.
"I hope that when we produce a framework document - that I think might surprise a few people, in terms of how open it is to quite fundamental change in a number of areas - I hope that perhaps some of these parties may change their mind and engage to put together a very ambitious programme for government".
Mr Varadkar said he is hopeful the document will be finalised this week.
"Well I can't speak for Fianna Fáil, but certainly we're ready to agree a framework document - but of course it takes two parties to agree.
"And then the intention is if we can agree that to approach other parties - like Labour, like the Greens, Social Democrats, perhaps groups of independents - to see if they're willing to form a new government.
"To help us through this emergency, to help us get our society and our economy going again - and also to look to the future".