However, the former rebel Maoists, who hold the greatest number of seats in parliament, are demanding that the government resign and allow them to lead a new coalition administration in return for the extension.
The Nepali Congress and the Communist Party of Nepal (UML), the other main political parties, have pushed for varied concessions from the Maoists in order to come to an agreement.
The UN urged for all sides on Thursday to come to an agreement in the interests of the peace process established in 2006 after a decade-long civil war.
"The constituent assembly and its progress to date toward the adoption of Nepal's new constitution represent a significant and hard-won achievement of the peace process," a UN statement said.
Ban Ki-moon, the UN secretary-general, appealed in the statement to the party leaders "to regain their unity of purpose in order to preserve the assembly and the peace".
He said now is the time "to put national interest first".
After changing from a monarchy to a republic in 2007, the Maoists won parliamentary elections in 2008. However, their rule lasted only nine months and they were left out of the succeeding coalition.
The current CA has failed to agree on the wording of a new constitution.
Yadav called for consensus during Thursday's meeting to prevent the country from falling into "political crisis", according to a presidential spokesman.
The Nepali Congress and UML were refusing to accede to Maoist demands to form a new government.
"The president asked us about the efforts being made to solve the prolonged political crisis," Jhalanath Khanal, the UML chairman, said after the meeting.
"In order to avert a constitutional crisis, which is likely following the failure to extend the CA, we all agreed that there is no alternative to consensus.
"We told him that we are trying to forge consensus."
Speaking to Al Jazeera, Yubaraj Ghimire, a Nepali political analyst based in Kathmandu, said: "The three political parties together make a two-thirds majority in the constituent assembly, which is a prerequisite for any constitutional amendment to be adopted.
"[They may say] at the last minute that they realise that 'we have a collective peril and we'll have to face the collective wrath of the people for not giving this constitution. So in order to survive six more months with these privileges, let's extend the term'.
"[But] I don't see any chance of that happening."
Ghimire said that even if the term was extended, there is no guarantee that the parties will deliver a constitution and end disagreements on as many as 20 issues, such as the model of federalism Nepal is to assume, independence of the judiciary and integrating Maoist combatants into the government.