The two parties left out on the cold are the opposition forces For Stability! and Latvia First.
Making the cut for a repeat invitation are New Unity, the Greens and Farmers Union (ZZS), the United List, the Progressives and the National Alliance – precisely the same parties that Prime Minister Krišjānis Kariņš wanted to form a "broader" coalition but which ultimately led to his resignation last week.
In a statement after the conclusion of Tuesday's round of talks, President Rinkēvičs said that he had received "fundamental support for the national development goals I have set and the urgent tasks to be carried out" from the five political forces asked back.
"Respecting the programmatic vision of the political forces and the activities of these forces in the parliament so far, I believe that the new Cabinet of Ministers needs to be formed with the broadest possible support of the majority of the Saeima. Such a government model would be optimal and actionable, providing concrete solutions to a number of important issues for the citizens of Latvia," he said, in what was largely a retread of his previous words on the subject.
"Along with the medium and long-term priorities I mentioned, the immediate task is to work on the 2024 state budget," he reasserted.
So far Evika Siliņa of the New Unity party – to which Rinkēvičs himself used to belong – is the only person put forward by any of the parties for the Prime Minister's position, but the president has yet to officially nominate anyone for the vacancy.
Wednesday's roundtable with represenatives of five different parties is likely to be the first serious test of Rinkēvičs' powers of persuasion, with longstanding antipathies between some of them in various different combinations.