Maj. Gen. Asif Ghafoor, Pakistan's chief military spokesperson, said in a tweet Wednesday that one Indian aircraft fell inside Pakistan-administered Kashmir, while another fell within the Indian-administered region of Kashmir.
Two Indian pilots have been arrested, Ghafoor later said, with one pilot said to be receiving treatment for injuries in a military hospital.
India did not say where its plane went down but a statement from Indian Foreign Ministry Spokesperson Raveesh Kumar, said an Indian Air Force plane (a MiG 21 Bison) shot down a Pakistani jet, which fell on the Pakistan side of the border.
Kumar said one of its pilots is missing and acknowledged that "Pakistan has claimed that he is in their custody."
Pakistan has claimed to have shot down two Indian jets and captured two pilots after a dogfight over Kashmir, igniting fears of an all-out conflict between the nuclear-armed neighbours.
Tensions remain high on the Asian Subcontinent where tens of thousands of Indian and Pakistani soldiers face off along the disputed Kashmir boundary.
There are competing claims regarding the exact details of what has taken place, but Pakistan's Major General Asif Ghafoor said one of the Indian pilots was in custody and the other in hospital.
Ghafoor said the jets had been shot down after Pakistani planes earlier Wednesday flew across the Line of Control, the de facto border in disputed Kashmir, to the Indian side in a show of strength, hitting non-military targets including supply depots.
Although this version of events is disputed by India, the Pakistani official said: "The Pakistan Air Force was ready, they took them on, there was an engagement. As a result both the Indian planes were shot down and the wreckage of one fell on our side while the wreckage of the other fell on their side."
Initially, the Indian Air Force (IAF) denied Pakistani claims, despite videos of the two pilots in Pakistani captivity being broadcast by state media. The IAF is also decried claims that two Indian fighter aircraft had been shot down.
But later on Wednesday a foreign ministry official told a press conference in Delhi that there was an "aerial engagement", conceding just one Indian jet was shot down.
IAF sources said that there were four Pakistani F-16 fighters against four IAF MiG-21 Bison combat aircraft and the dogfight happened in a chase.
The four Pakistani fighters are believed to have tried attacking an ammunition dump at Nowshera near the Line of Control in Kashmir, when they were chased by four Indian planes.
India claims it has also shot down one of the Pakistani fighter jets.
The incident is the latest in a dangerous sequence of events between the two countries, whose ties have been under intense strain since a February 14 suicide bombing in Indian Kashmir that killed 40 troops.