New Delhi has accused Islamabad of supporting the perpetrators of the February 14 attack that left at least 44 paramilitaries dead. However, Pakistan has categorically denied any involvement.
Two Pakistani shooters, GM Bashir and Khalil Ahmed, and their manager were to take part in the event but had not arrived for Wednesday's eve-of-contest formalities.
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The International Shooting Sport Federation (ISSF) warned India of consequences.
"The ISSF World Cup in New Delhi faces an urgent situation as Pakistani athletes cannot get entry visas to participate in the competition," it said in a statement.
The federation and the Indian organising committee are making "all efforts to solve the situation to avoid the discrimination of the Pakistani team".
"Besides that, the ISSF and the organizing committee are discussing the possible consequences for India as a host country for future international competitions, in all sports."
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India, which has expressed a strong interest in hosting the 2032 Olympics and 2030 Asian Games, could face International Olympic Committee (IOC) action if Pakistan is blocked from taking part.
The event, with 500 shooters from 58 nations, is a qualifier for the 2020 Tokyo Olympics.
National Rifle Association of India general secretary D.V. Seetharama Rao said it was not certain the Pakistanis would cross the border even if they had visas.
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"They have to come and shoot and shooting is such a technical sport with so much mental strain that this might cause disturbance in their mind," Rao told reporters. "Their families and coaches might not allow them to come."
The official added: "Repercussions? we take it as it comes. I can't predict the future."
Rao also said the Pakistanis would be allowed to compete if they have visas and make the trip to India.
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