Trump, speaking at a press conference in Hanoi after a summit with North Korean leader Kim Jong Un, said he had had "reasonably attractive news from Pakistan and India".

The US has been "involved in trying to have them stop," he told reporters.

"We have some reasonably decent news... Hopefully that's going to be coming to an end," he added.

The comments were his first since India and Pakistan both claimed to have shot down each other's fighter planes after a rare aerial engagement in the skies over the disputed Himalayan region of Kashmir early Wednesday, with Pakistan capturing one Indian pilot.

The incident -- the latest in a dangerous sequence of events between the two countries -- sent tensions rocketing, as major world powers including China, the US and the UN urged restraint.

New Delhi said it lost one fighter jet and a pilot was missing in action, later confirming he was in Pakistani custody. India also downed a Pakistani jet, the foreign ministry said.

Pakistan claimed to have shot down two Indian jets and denied it lost any aircraft in the dogfight.

Islamabad said its incursion across the heavily militarised border was in response to Indian warplanes bombing Balakot, well inside its territory, on February 26.

That attack was the "jaw breaking" response Indian Prime Minister Narendra Modi had promised after militants staged a suicide bombing in Indian Kashmir on February 14, killing at least 40 paramilitaries.

New Delhi blamed the Pakistan-based Jaish-e-Mohammad for the attack, and hit what it called a training camp for the militants in its aerial raid 12 days later.

The crisis has seen the US thrust into the familiar position of trying to defuse tensions, but American officials are making no secret of faulting Islamabad.

US policymakers have grown exasperated with Pakistan for not doing more against Islamist extremists, and envision a broad, long-term alliance with an emerging India.