Mary Lou McDonald has spoken to Micheál Martin by phone, as both Sinn Féin and Fianna Fáil continue to consider their government formation options.
The Fianna Fáil parliamentary party ruled out entering a coalition with Sinn Féin on Thursday, meaning Mr Martin would have to seek an arrangement with Fine Gael.
However, Ms McDonald says a 'grand coalition' would be "precisely what [people] voted against" in last week's election.
Following a 15-minute phone call with the Fianna Fáil leader on Friday afternoon, Ms McDonald said leaders have a duty to engage with others.
She said: "I told Micheál Martin that people who voted for Sinn Féin and those who voted more widely for change are angry that Fianna Fáil is denying them the respect of sitting down with the party that represents them.
"I will write again to Micheál Martin in the coming days to set out the substance of our position and have asked him to do the same.
"Any meeting between us should be about building homes, giving workers and families a break, advancing Irish unity and delivering on the desire of the people for a new Government for Change."
Mr Martin said he congratulated Sinn Fein on their election success, but reiterated the reasons why Fianna Fáil were not going into coalition with them.
He said: "These were economic and about issues of the past and current structures.
"Fianna Fáil has a mandate and that it is our prerogative and duty to try to form a government.
"No one grouping in the Dail can force another grouping to form a government when there are such incompatible views on economic policy and other issues."
Earlier on Friday, Ms McDonald sharply criticised the "old boys club" of Fianna Fáil and Fine Gael, claiming they are not respecting what the electorate voted for.
Meanwhile, Solidarity's Mick Barry believes the numbers are not there for a left-wing government.
He said now is the time for an opposition movement to be created in the event Fianna Fáil and Fine Gael come to an arrangement for a new government - saying it should be organised while government talks are ongoing.
Mr Barry also called on Sinn Féin and the Greens to rule out a coalition with Fianna Fáil and Fine Gael.
Representatives of People Before Profit and RISE had been due to meet with the Green Party on Friday, but that meeting has been postponed until early next week.
Additional reporting: Jack Quann