"We insist that the acting prime minister, the speaker of the parliament and the head of the parliamentary majority participate in the negotiations", Pashinyan at a briefing in Yerevan.
It was not immediately clear whether Sargsyan's exit will be followed by fresh parliamentary elections.
"If people will bestow on me this responsibility, I'll assume the responsibility", Pashinyan said.
Serzh Sargsyan, Armenia's former president who lawmakers voted in as prime minister just last Tuesday, announced on Monday that he will step down.
After the latest political events in Armenia, where the Prime Minister Serzh Sargsyan has resigned after the public dissatisfaction, the OSCE Minsk Group and the United Nations have issued statements where the organizations call on Azerbaijan and Armenia to avoid escalation in the region.
Karapetyan said on Tuesday that "the current impasse can only be resolved through political negotiations and while we respect the wishes and freedom of the people on the streets, we need to adhere to the country's laws and constitution".
But following a referendum in 2015, Armenia transitioned from an executive presidency with Sargsyan at the helm to a parliamentary system led by the prime minister and his Cabinet.
He was first elected president in a hotly disputed 2008 ballot marred by opposition allegations of fraud and a deadly post-election government crackdown on protesters in Yerevan.
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Pashinyan was going to meet the acting PM of Armenia, Karen Karapetyan, but the Armenian media informs the meeting has been cancelled.
The new constitution states that the president serves for seven years and can not be elected to consecutive terms.
However, the second biggest party in parliament said on Wednesday it was joining the protest movement and would encourage its supporters to take to the streets.
The Kremlin said on Tuesday it did not regard political tumult in Armenia, a close Russian ally, as a Ukraine-style revolution and was pleased that the situation there appeared stable for now.
"As a friend and partner to Armenia, we commend the Armenian people for engaging in dialogue to forge their sovereign future through democratic and peaceful means", the statement concludes. The ministry said Russian Federation is following developments in Armenia closely and wishes the country a smooth and peaceful political transition.
The new prime minister must be a "people's candidate" and not a member of Sarkisian's ruling Republican Party, he stressed.
"We welcome the courageous decision of Prime Minister Sargsyan to resign from his post in order to defuse the mounting tensions in the country".
In his address in Republic Square on Monday, Pashinyan declared "victory" and said "the revolution must not stop with Sargsyan's resignation, but continue to final victory".