Hope is growing in Fine Gael and Fianna Fail that they can put together proposals for government that will entice at least one smaller party into a coalition.
Senior politicians from both parties met again last night as they thrashed out a framework document on tackling coronavirus and pre-existing issues in housing and health.
The plan is to present the document to the Green Party, Social Democrats and Labour.
Taoiseach Leo Varadkar said such parties shouldn't be bullied into government and said he doesn't see a second election taking place.
Tanaiste Simon Coveney said he hoped the framework document might surprise some people in terms of the change it offered.
A senior Fianna Fail source said progress continued to be made with Fine Gael and the party "remains hopeful smaller parties will engage and ultimately participate in producing a programme for government".
Negotiating teams from Fine Gael and Fianna Fail are to meet again tomorrow or Thursday as they continue to finalise the framework document.
Both parties have said they want a solid Dail majority but Fine Gael and Fianna Fail have only 72 seats of the required bare minimum of 80.
The Greens, Social Democrats and Labour, having chosen Alan Kelly as their new party leader, have all expressed reluctance to enter coalition with the two larger parties and Fine Gael and Fianna Fail have ruled out a deal with Sinn Fein.
Failure to get a small party on board would leave the potential government relying on several Independent TDs to elect a Taoiseach.
Mr Varadkar last night said that if the two larger parties agreed a framework for government it would be presented to the smaller parties with an understanding that they would want key parts of their agendas implemented.
"What we're saying really to those parties is... we wouldn't expect for a second that you would enter government unless a big part of your core agenda was a part of that government's mission as well," he said.