Jhala Nath Khanal-led CPN-UML, which emerged as the second largest party in the November 19 elections, said the appointment of the president should be sorted out according to the people's fresh mandate.
Nepali Congress (NC), which emerged as the largest party in the polls, should present a consensus proposal, CPN-UML secretary Shanker Pokharel told reporters.
The NC should incorporate all the issues, including the government, president and drafting of a new constitution, in the proposal, he said.
Meanwhile, NC leader Prakash Sharan Mahat said his party wants to form a government and draft constitution with a consensus in order to draft the new constitution.
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"Constitution can not be drafted with a system of 100 per cent consensus," Mahat said.
The pro-monarchist Rastriya Prajatantra Party of Nepal also echoed CPN-UML's view on a consensus before choosing a new president.
The party is likely to get 25 seats out of total 335 seats to be allotted under the proportional system.
"There should be new election of president, vice-president and other elected posts as per the fresh mandate of the people, and political consensus should be sought among parties keeping in view the new power equations" RPP Nepal president Kamal Thapa said after his party emerged as the fourth largest in proportional vote count.