It said the Sena will take a "final decision" on Kovind in the meeting of its party leaders scheduled here tomorrow.
Addressing the 51st Foundation Day function of Sena in suburban Matunga this evening, Thackeray said his party will not back Kovind if his candidature is aimed at garnering Dalit votes.
Thackeray had reportedly told visiting BJP chief Amit Shah yesterday that the Sena would take a call on supporting the NDA nominee for the top constitutional post only after BJP reveals its choice. Shah had called on Thackeray at his residence here yesterday.
"If the name of Kovind is being put forward with a view to garner dalit votes then the Sena is not interested.
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Sena has never indulged in the politics of hiding behind someone. We shall decide on whether or not to support the candidature of Kovind tomorrow," Thackeray said.
Sena had earlier rooted for the candidature of RSS chief Mohan Bhagwat on Hindutva line and later proposed the name of veteran agriculturist M S Swaminathan for the top post.
Sena, known for its unpredictability, backed the UPA nominee Pranab Mukherjee in the 2012 presidential polls against the NDA's P A Sangma. In 2007, Sena backed Pratibha Patil instead of NDA candidate Bhairon Singh Shekhawat.
Thackeray said Sena had proposed Bhagwat's name while keeping in mind the concept of 'Hindurashtra' (Hindu nation).
"Considering that there may be objections to Bhagwat's name, we mooted the name of Dr M S Swaminathan taking note of his service to the cause of farmers," he told the gathering of Shiv Sainiks.
Earlier in the day, Sena spokesperson Sanjay Raut said Shah called up Thackeray after the NDA's presidential choice was decided by the BJP's parliamentary board meeting, and sought the party's support for Kovind.
In the 288-member Maharashtra Assembly, the BJP has 122 MLAs and claims to have support of some more MLAs. Sena has 63 MLAs.
There are 67 members from Maharashtra in Parliament - 48 in the Lok Sabha and 19 in the Rajya Sabha.
The BJP-led NDA (including Sena) has total of 52 MPs.
Sena has 21 MPs - 18 in Lok Sabha and 3 in Rajya Sabha, while the BJP has 23 members in Lok Sabha and 5 members in Rajya Sabha from the state, that corresponds to a total value of 19,824 votes.
During his address, Thackeray took potshots at Chief Minister Devendra Fadnavis over his remarks that some political parties were behind the recent agitation by farmers.
"...Sena not only stood behind farmers just in spirit, but also with full might," he said.
Against the backdrop of a speculation that mid-term polls could be in offing in Maharashtra, he said Sena is not unduly worried about the snap polls, but is more concerned about suicides being committed by farmers.
"Nobody should think that they can keep on winning elections all the time. If anyone is so boisterous then let them hold mid-term elections. We shall bury you in the soil of Maharashtra," Thackeray said in an apparent dig at Amit Shah over his recent statement that BJP is winning elections in various states.
Thackeray said the state government should not put stringent conditions upon farmers to avail the farm loan waiver, which was announced recently.