Mr Vajpayee addressed Kashmiris on Friday
India has said Pakistan could open the door to talks over Kashmir if it ended militant incursions in the disputed region.
In a statement to the Indian parliament, Prime Minister Atal Behari Vajpayee repeated an offer made last week while on a visit to Kashmir.
Pakistan President Pervez Musharraf has responded to the offer by saying he is ready for a comprehensive dialogue on all outstanding disputes.
Both sides should resolve that we need to live together in peace
Prime Minister Atal Behari Vajpayee
Despite the apparent thaw in relations, at least 25 people died in a day of violence on Tuesday in Indian-administered Kashmir.
Mr Vajpayee said on Wednesday that Islamabad could pave the way for talks by "stopping cross-border infiltration and [the] destruction of terrorist infrastructure".
"I said [in Kashmir] that we have extended our hand of friendship. Let us see how Pakistan responds to this.
"Both sides should resolve that we need to live together in peace," Mr Vajpayee said.
Pakistan has always denied giving the militants anything more than moral support in their struggle against Indian rule.
In his response President Musharraf welcomed India's "positive indications".
"Pakistan wishes to resolve all its disputes with India peacefully," he said
On Tuesday, Pakistan's foreign office said it was ready to nominate a senior official to discuss the agenda for talks.
"We believe the only way forward is through sincere, unconditional, meaningful and sustained dialogue addressing the key issues bedeviling relations between the two countries," it said in a statement.
But correspondent says it is too early to say whether the public statements will lead to an early resumption of talks.
India and Pakistan have fought two of their three wars since Independence over Kashmir.
In a landmark public address in Srinagar on Friday, Mr Vajpayee had said that dialogue was the only way to bring peace to Kashmir.
The rally was the first in the Muslim-dominated Kashmir valley by an Indian prime minister in 16 years.
"We have come to share your pain and grief," the prime minister told a crowd of 20,000.
Mr Vajpayee also promised to provide more jobs and assistance for the Kashmiri people and said he was fully behind the state's new government.
The new administration of Mufti Mohammad Sayeed won office by promising to fight separatism by improving living conditions.