The Armenian Revolutionary Federation, also known as Dashnaktsutyun, said after leaving the governing coalition that the parliament should elect a prime minister who "enjoys the people's confidence".
Sargsyan, 63, who was elected Armenian prime minister by the parliament on April 17, said in a statement on Monday that he would resign following days of street protests against him.
Protest leader and parliament member, Nikol Pashinyan, whose liberal Way Out Alliance comprises a small minority in the legislature, is likely to be a candidate.
The shift to a strengthened parliamentary triggered massive anti-government protests in the capital of Yerevan on April 13, with thousands of people participating in rolling demonstrations against Sargsyan.
He did not say when new elections might be held.
"But let me clearly stress that my becoming prime minister isn't the issue, but rather the removal of the corrupt and anti-people regime".
Pashinyan demands 'unconditional capitulation' of Armenia's ruling party as protests continue
If elected, he wants to reform the electoral system to ensure it is fair before holding new parliamentary elections.
Keran Karapetyan, the acting prime minister, has said he was ready for elections, but not on the street.
The protests were sparked by Sargsyan's move to extend his rule.
Thus, what started as street protests in Yerevan by the opposition parties to register their displeasure for the country's economic problems, corruption, and monopolistic control of economy and politics by the oligarchs, quickly turned into a massive rally with the participation of other groups and finally, unarmed Armenian soldiers.
Last Saturday, after a meeting between Sargsian and Pashinian that was televised, during which Pashinian stood firm saying: "The only thing there is to negotiate is your resignation", the Prime Minister abruptly withdrew at the end of the meeting.
The Armenian government's press service said Gevorgian met with representatives of Russian President Vladimir Putin's administration.