Though Prime Minister Narendra Modi informed Congress chief Sonia Gandhi personally about the BJP's candidate for President of India, her party said that it was a token gesture.
"They only revealed the name of their choice after making the decision. So, it was simply their decision, then what's the point of holding talks?" asked Congress leader Ghulam Nabi Azad.
"Opposition parties will meet on June 22 (Thursday) at 4:30 pm to come to a common stand," said Left leader Sitaram Yechury. He also repeated the Congress' charge - that though the BJP promised consultation, what it has sought is cooperation.
Opposition leaders had said they would wait for the government's nominee before deciding whether to put up a contender of their own.
Governor Kovind, trained as a lawyer and the son of poor farmers, also used to lead a BJP unit that represents Dalits. His caste could make it hard for many parties to oppose his promotion to President.
The BJP's traditional support base is among upper-caste Hindus but the party has sought to broaden its appeal, winning a landslide victory in Uttar Pradesh in March with the backing of minority voters.
The BJP had assigned three senior ministers to talk to allies and the opposition last week. Over the weekend, Mr Shah met with Sena president Uddhav Thackeray, who regularly and aggressively asserts his differences with the BJP despite serving as its ally in both Maharashtra and the national ruling coalition.
"We were not given the name earlier, we have been given it only now," complained Sena leader Sanjay Raut.
Nominations for President close on June 28 and an election, should one be required, will be held on July 17.
Under a complex electoral formula, the BJP and its allies control about 48 per cent of votes in the Electoral College but should have enough support from other parties like the AIADMK to secure a majority.