Members of the Congress Party and the regional People's Democratic Party met separately in the state capital, Srinagar, on Monday after a strong showing in the polls.
Both teams emerged from their talks insisting that their leaders be appointed chief minister, with no sign of a willingness to compromise.
Meanwhile, the defeated National Conference leader has announced his resignation as India's junior foreign minister, saying he wanted to rebuild his party.
Omar Abdullah said: ''I resigned because I have to work in Jammu and Kashmir to strengthen the party there. I am needed more in the state in view of the election results.''
The National Conference is still the largest party in Indian-administered Kashmir despite a disastrous perfomance in which it lost 29 of its 57 seats.
Congress won 20 seats and the PDP 16 in the 87-member assembly.
Governor's talks
Kashmir Governor Gary Saxena held exploratory consultations on forming a government with both Congress and the PDP on Monday. He also met Mr Abdullah.
Congress has put forward Ghulam Nabi Azad as its candidate for chief minister by electing him its state party chief.
But the PDP insists its leader, Mufti Mohammad Sayeed, should head the government.
He said on Monday that the PDP differed with Congress on fundamental issues.
''Congress is not ready to hold unconditional dialogue with separatists and we should reach a minimum agreement before we go ahead with the alliance,'' Mr Sayeed said.
The PDP wants the top post to go to a Muslim from the Kashmir Valley, the centre of the armed insurgency which has left tens of thousands dead in the past decade.
Mr Azad is a Muslim from the Hindu-dominated Jammu region.
He says he will seek agreement with the PDP and other "like-minded parties and individuals", but would still try to form a government if he fails.
"Congress is of the opinion that it should head the government but the PDP wants the lead role," he said.
Conference hopes
Mr Abdullah lost his own seat and his party surrendered power for the first time in almost three decades.
The National Conference could still stitch together an alternative coalition to retain power, although analysts believe they will not find enough allies.
Whoever tries to form a government will have to rely on the support of smaller parties and independents who have more than 20 seats.
"We are still in the race [to form a government] and people are joining us as we go along," said Farooq Abdullah, the state's outgoing chief minister and Omar Abdullah's father.