"Let me be clear, though: our vision for the Indo-Pacific excludes no nation".
In a readout, Pence's office said he encouraged "free, fair, and reciprocal trade with India".
But India remains wary of being drawn into any U.S. strategy that might involve the containment of China, although the two nations have stepped up bilateral defence ties and military exercises.
"There was some good understanding of the areas of how we move forward in building cooperation on counter-terrorism and both countries recognised this is a challenge which we have to fight together and to fight along with the rest of the worldwide community", Gokhale said.
Trump has said he hopes to have a second meeting with Kim early next year, but talks between Secretary of State Mike Pompeo and a top North Korean official planned for last week, partly to prepare for the meeting, were cancelled.
South Korean President Moon Jae-in and his Russian counterpart, Vladimir Putin, talked about easing sanctions on North Korea in their talks Wednesday on the sidelines of a regional summit in Singapore, Moon's office said.
PM Modi also reminded the U.S. vice-president that in one way or another all the traces and all the leads in global terror attacks ultimately lead to a "single source and single place of origin", in an apparent reference to Pakistan.
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The vice president also highlighted the United States' "pressure campaign" on North Korea, its "commitment to uphold the freedom of the seas and skies" and determination to ensure that Southeast Asian nations are secure in their sovereign borders, on land, and at sea in the digital world. The two also discussed enhancement in defence ties with India having acquired billions of dollars of United States defence platforms, equipment and weaponry in the past decade. They also discussed counter-terrorism cooperation and ways to boost trade, with India indicating interest in expanding the newly initiated imports of oil and gas from the U.S., now valued at US$4 billion (S$5.5 billion).
Mr Modi also asked the USA to explore making defence equipment in India not just for its own market but with an eye on regional markets. The country claims most of the waterway, through which about US$5 trillion in seaborne trade passes each year.
"So there was some good understanding on the area of how we move forward in building cooperation on counter-terrorism and both countries recognised that this is a challenge which we will have to fight together and to fight along with the rest of the worldwide community", he said.
Pence acknowledged that India had made substantial progress economically as well as diplomatically in regional and worldwide affairs, Gokhale said.
"Prime Minister referred to his speech here in Singapore in June at the Shangri-La Dialogue in which he had outlined India's vision of the Indo-Pacific", Gokhale said. "We are happy that ties with ASEAN are strong and are contributing to a peaceful and prosperous planet", Modi tweeted.
Pence thanked India in supporting the USA on its North Korean pressure campaign.