Speaking to reporters after the 30-minute meeting held at the CPI(M) head office in New Delhi, Mr Yechury reiterated his party's stand that a person with "impeccable secular credentials" should be the next president.
The CPI(M) leader said the opposition parties would wait till the government completed the round of discussions with them to take a call on their nominee for the July 17 presidential election.
According to sources in the CPI(M), the Left party wanted to know from the ministers, who are part of a committee the BJP has floated to evolve consensus on the presidential nominee, if former Delhi Metro chief E Sreedharan was the NDA's candidate, as reported by a website.
The ministers, the sources said, denied the report and told Mr Yechury that the government was yet to finalise any name.
A senior CPI(M) leader wondered if the RSS-BJP combine would let go the opportunity to install a president of their choice since "they are so close to" electing one to the post.
"They told us they want our support. We said fine, tell us the bases for the support, who is the candidate. But that they did not convey... They will take some time to do that," Mr Yechury told reporters after the meeting.
CPI(M) Politburo members Prakash Karat and Brinda Karat also attended the meeting.
Mr Yechury said it was required for the government to suggest names for arriving at a consensus.
With no name proposed by the government, Mr Yechury said, the parleys looked "more like a public relation exercise" of meeting opposition parties.
"Since the government is doing it, we welcomed them in our office. Very good, they came here," he said.
Countering the opinion that the opposition should support the ruling alliance's candidate since it has the people's mandate, Mr Yechury said the NDA had polled 39 per cent votes during the 2014 general polls as against 61 per cent of the opposition.
"So, a candidate of the opposition is the one who has the mandate. So, the government should also consider that aspect," he said.
Mr Yechury insisted the Centre should also be open to considering the names suggested by the opposition if it made any proposal.