"We want him back", sources said. "It is good for both sides".
Tensions between the archrivals escalated after a suicide attack in Indian-administered Kashmir killed at least 42 Indian soldiers.
Pakistan arrested the Indian pilot after aircraft from the two nations clashed in the disputed region of Kashmir in the worst military stand-off in decades.
Meanwhile, members of Modi's Bharitiya Janata Party called for more military action, suggesting the conflict still could worsen.
Modi is running for reelection later in the spring and has staked his reputation on a muscular approach to national security.
Thousands of air travellers stranded worldwide heaved a sigh of relief as Pakistan announced that its airspace will be fully re-opened by Friday, as the aviation authorities allowed some flights to operate Thursday a day after closing the air traffic over the country in view of escalating tensions with India.
At the same time, he said the lack of USA condemnation of India's strike on Pakistan was "construed and understood as an endorsement of the Indian position, and that is what emboldened them even more".
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According to the details, the footages had been taken by Pakistan Air Force aircraft and shared on YouTube three years ago. Fearing more attacks, hundreds of civilians on Thursday fled villages on the Pakistani side of the border.
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Government buildings in Muzafarabad, the capital of the Pakistan-controlled section of Kashmir, were used to provide shelter to those who fled from border towns.
India shared evidence that a Pakistani F-16 fighter had been shot down in aerial combat on Wednesday but has not yet made public evidence of the damage inflicted on the JeM training camp. Pakistan's airspace remained closed for a second day Thursday, snarling air traffic.
Mr Trump, speaking at a press conference in Hanoi yesterday after a summit with North Korean leader Kim Jong Un, said he had had "reasonably attractive news from Pakistan and India".
"I think hopefully that's going to be coming to an end", Trump said, without elaborating.
"As a country with close and abiding relations with both countries, we encourage them to exercise maximum restraint, and to re-engage in dialogue and negotiations". Modi's opponents seized on his decision to continue election-related activities in the midst of the crisis and criticized him for saying nothing about the pilot's capture. "The country is facing challenging times and it will fight, live, work and win unitedly".
Yet the release of the pilot, named as Wing Commander Abhinandan Varthaman, may be just enough for both countries' governments to appear victorious in the eyes of their people and stop the escalation.