Sources said Singh and Naidu - the panel's third member, Arun Jaitley, is overseas - told the opposition leaders that the party would like to finalise the name of the candidate for the post before Modi left on a foreign tour on June 24. Finance minister Arun Jaitley is returning from overseas on Saturday and is expected to touch base with socialist parties over the next few days.
Stepping out from the almost 30-minute meeting, Yechury told media persons that the BJP panel would meet them again.
However, government sources said disclosing the name of its candidate before consultations with political parties may not be appropriate.
Senior Congress leader Mallikarjun Kharge later told reporters that there was no question of consensus "when they have not given any names".
"They told us they want our support".
"We have nearly 3 lakh votes more than the required number in the presidential electoral college", said a senior leader referring to the additional support the BJP has managed to get from parties outside the NDA.
Union ministers Rajnath Singh and M Venkaiah Naidu, two members of the panel formed by BJP chief Amit Shah, met Congress chief Sonia Gandhi, CPI (M) general secretary Sitaram Yechury and BJP veterans Advani and Murli Manohar Joshi.
The key question thus remains who would be ruling NDA's candidate as proposed by Modi in due course.More news: Xbox One X will not generate profits for Microsoft
The first meeting of the day was with Ms. Gandhi and her colleagues - Leader of Opposition in the Rajya Sabha Ghulam Nabi Azad and Congress Leader in the Lok Sabha Mallikarjuna Kharge. "Since the BJP panel of three ministers has consulted the wide spectrum of the political class on evolving consensus, it's now time to consult own allies", said a senior BJP functionary. Arch rivals in West Bengal's politics, Trinamool Congress and CPM, had also supported Gandhi's name.
With no name proposed by the government, Yechury said, the parleys looked "more like a public relation exercise" of meeting opposition parties. One name also doing the rounds is that of Vijay Pandurang Bhatkar, current Chancellor of Nalanda University, a technocrat who is considered the father of the Indian Supercomputer and president of the RSS-affiliated Vijnana Bharati, which promotes "swadeshi" science. We will think about it when they suggest a name. So, it was expected they (government) will suggest some names.
The opposition leaders were told that the government and the BJP would like to finalise the candidate for the presidential poll before Prime Minister Narendra Modi leaves on a three-nation trip on June 24. June 22, they say is the most appropriate date to make that big announcement.
Naidu also had a conversation on this with NCP Chief Sharad Pawar over the phone and met AIADMK leader Thambidurai.
Meanwhile, Rashtriya Swayamsevak Sangh (RSS) chief Mohan Bhagwat had lunch with President Pranab Mukherjee, a meeting the RSS described as "a pre-decided courtesy meeting" into which "nothing should be read".
In the same breath, Sena President Uddhav Thackeray also said that if the BJP President has a good candidate, his party (Sena) would give it a thought. Nevertheless, the party had undertaken an exercise to know the mind of the opposition leaders, he said. While Rahul Gandhi, 46, has been a singularly dismal leader for the party with his curriculum vitae a blank space sans a 2009 Uttar Pradesh Lok Sabha win credited to him, he will take over at a time when the party is on the verge of a split in Gujarat, which will go to the polls in six months.
Stating that the party leaders were not aware of any such proposal, the source said it would have been in the fitness of things had the BJP first sent a formal proposal.
Backing his stance, Congress leader Tom Vadakkan said it is the decision of the Government hereafter, adding that the representatives at the meeting were part of a "fact-finding committee".