The president of India's main opposition Bharatiya Janata Party (BJP) has resigned, citing personal reasons, after its latest electoral defeat.
Naidu (L) made the offer at Vajpayee's residence on Monday
Venkaiah Naidu quit after a coalition led by the party was defeated in the state election in Maharashtra.
Former Indian deputy prime minister LK Advani was appointed the new president after talks with party chiefs in Delhi.
It will be Mr Advani's third term as party president. He already acts as the leader of the opposition in parliament.
His appointment was unanimously backed by BJP leaders, including the former prime minister, Atal Behari Vajpayee.
Mr Naidu told a news conference in Delhi he was happy that his resignation had been accepted by the party.
Mr Naidu said he was quitting because of personal reasons. He said his wife was very ill and needed his time.
He said: "I felt I should make way for a senior in the party."
There were many challenges before the party and the nation, he added, saying he wanted a senior colleague to take over as the party president, to study the challenges and take the party and the country forward.
BJP leaders unanimously backed Advani's appointment
Mr Advani is considered one of the more hawkish members of the Hindu nationalist BJP.
The BBC's Jyotsna Singh in Delhi says the key issue under his leadership will be whether the party pursues its strident Hindu nationalist line or broadens the party's base by reaching out to minorities, particularly Muslims.
Meanwhile, the Congress-Nationalist Congress Party, which won the poll, is still deadlocked over who to appoint as chief minister.
The formation of the government may be delayed a few days.
The Nationalist Congress Party (NCP), which won most seats, wants to choose the leader rather than its ally, the Congress Party.
State Congress leaders have left for Delhi to consult leader Sonia Gandhi.
Sharad Pawar, the NCP leader, said after talks with Mrs Gandhi in Delhi that the alliance had agreed earlier on who would become chief minister.
Sonia Gandhi and Sharad Pawar must resolve the leadership issue
"As the formula has been in force for five years after it was agreed upon, that formula has to continue," he said. "That is what we want to say."
The NCP won 71 seats to Congress' 69 and so is the senior coalition partner.
However, Congress spokesman Sanjay Dutt said: "Our party workers strongly feel that the chief minister
should be from Congress."
State Congress leaders say they will leave a decision to Mrs Gandhi.
The BBC's Zubair Ahmed in Mumbai (Bombay) says there are reports that current chief minister, Sushil Kumar Shinde, who represents Congress, may emerge as a compromise candidate.
The alliance was still a few seats short of the 145 it needs for a majority in the state assembly, and requires the support of minor parties to govern.
The poll was seen as a key test of Congress popularity following its surprise general election win in May.
The state is India's most industrially advanced and considered a major political prize.