"The Sena will have to inform the governor about its stand by Monday (November 11) evening 7.30 pm," a Raj Bhavan official said.
In the 288-member Assembly, the BJP and the Sena each has 105 and 56 MLAs, respectively.
Opposition Congress and NCP won 44 and 54 seats, respectively, in recent assembly polls. The halfway mark is 145.
Koshyariasked Sena legislative party leader Eknath Shinde to "indicate willingness and ability" of the party to form government in Maharashtra, a Raj Bhavan statement said.
The governor had on Saturday invited the BJP, the single largest party, to stake claim to form government.
However, with the BJP unable to muster numbers in view of the Sena's stand not to support it on the issue of sharing of power and the post of the chief minister, the party on Sunday declined to form government.
"The Governor has, therefore, asked the leader of elected members of the second largest party, Shiv Sena, Eknath Shinde, to convey its willingness and ability to form the government," the statement said.
With the BJP backing off from the race, the onus to form government now rests with the Sena, which has indicated that it might rope in the Congress and the NCP to prop up a government.
Sena MP Sanjay Raut had earlier claimed support of 170 MLAs.