Taoiseach Leo Varadkar and Fianna Fáil leader Micheál Martin will hold exploratory talks on Tuesday to determine if there is a basis for a historic first coalition between the two parties which have dominated Irish politics for a century.

With Fine Gael signalling a growing willingness over the weekend to re-examine its preference to go into opposition, Mr Varadkar is expected to report back to his parliamentary party on Wednesday if there is a basis for pursuing matters further with Mr Martin.

The meeting takes place in a week in which all parties will intensify engagements to explore government formation. Sinn Féin and the Green Party will begin two days of detailed policy discussions on Monday.

Sinn Féin has also announced a series of public rallies around the State over the next fortnight. It is hoping large attendances will increase public pressure on the other two big parties to reverse their refusal to enter talks with it.

Separately, Green Party leader Eamon Ryan has said his party will hold similar two-day policy discussions with Fianna Fáil later this week, and Fine Gael early next week. The party is reluctant to go into any coalition with two bigger parties alone, and wants other smaller parties as participants.

As things stand Mr Varadkar has no mandate from Fine Gael to negotiate a formal programme for government, and the party’s formal position is that it is preparing for opposition, with the onus falling on Fianna Fáil and Sinn Féin to form a government.

However, Garda Commissioner Drew Harris’s comments on Friday – where he said he agreed with a PSNI assessment from 2015 of a remaining IRA influence on Sinn Féin – have reinforced Mr Martin’s stance not to to enter any coalition talks with that party’s president Mary Lou McDonald. Until Mr Harris’s comments some Fianna Fáil TDs had privately questioned if Mr Martin’s trenchant exclusion of Sinn Féin, plus his willingness to enter talks with Fine Gael, was the correct strategy for the party.

Within senior levels of both parties there is now a growing sense that a coalition involving them is the only remaining pathway to a stable majority government.

A Fine Gael spokesman said its mandate as one of three main parties would have to be respected in the composition of any programme for government and composition of Cabinet.

Sources said Fine Gael could demand a rotating taoiseach and an acceptance by Fianna Fáil it is not the lead party.

Very arrogant

On Sunday, Fianna Fáil TD Robert Troy said his party should be considered the lead party but this was described by one senior Fine Gael source as “very arrogant”.

“It’s hard to see any basis for meaningful talks this week if Mr Martin takes a similar approach when he meets the Taoiseach,” said the source.

On Sunday Sinn Féin repeated its strong denial that the Provisional IRA retained any influence over Sinn Féin.“The IRA has gone, and they are not coming back. That’s the reality,” Donegal TD Pádraig Mac Lochlainn told RTÉ’s The Week in Politics.

Deputy leader Michelle O’Neill said Mr Harris’s comments had been manipulated by Fine Gael and Fianna Fáil.“It comes at a time when they are shaping up to do more of the same together in an attempt to carve up political power for themselves.”

Speaking of the rallies, Mr Mac Lochlainn said: “[The wishes of] 700,000 voters are not being respected. To say we are not fit for government is not respecting our mandate.”