Briton Tom Ballard and Italian Daniele Nardi last made contact from Nanga Parbat at an altitude of about 20,700ft (6,300m) almost two weeks ago.
On Wednesday it was reported the search had been called off, but resumed when “silhouettes” were spotted on a passage taken by the climbers.
Officials have now confirmed the two “shapes” are the missing men.
Stefano Pontecorvo, the Italian ambassador to Pakistan, said Spanish climber Alex Txikon found the bodies on the Mummery Spur trail.
Mr Ballard, 30, originally from Belper in Derbyshire, is the son of Alison Hargreaves, who died descending from the summit of K2 in 1995 – the same year she became the first woman to conquer Everest unaided.
Ahead of her death, he had moved to Fort William in Lochaber in the Scottish Highlands with his sister Kate and father Jim.
Mr Ballard and Mr Nardi, 42, last made contact with their team at base camp on 24 February as they tried to reach the summit of Nanga Parbat – the world’s ninth highest mountain.
A number of deaths on the peak, which is notoriously difficult to climb, have earned it the nickname “Killer Mountain”.
Mr Pontecorvo said the bodies were in a place that was difficult to reach but everything possible would be done to try and recover them.
Confirming the news on his official Facebook page, Mr Nardi’s team wrote: “We are devastated by pain; we inform you that Daniele and Tom’s searches are completed.
“Part of them will remain forever at Nanga Parbat.”
They said Mr Nardi was a “lover of life and adventures, scrupulous, brave, loyal, attentive to details and always present in moments of need”.
The statement added: “The family remembers Tom as a competent alpinist and brave friend of Daniele. Our thoughts are with him.”
Writing on Facebook, Mr Ballard’s girlfriend Stefania Pederiva said her heart was “completely drowned”.
“There are or will never be words suitable to describe the void you left,” she added.
“I thank the universe for giving me such a special person, there are only the wonderful memories of the times spent together that are the most beautiful of my life.”
Searches for the men began days after they last made contact with their team, but these were delayed because of bad weather and tensions between Pakistan and India.
Mr Nardi, from near Rome, had attempted the Nanga Parbat summit in winter several times in the past.
In 2015, Mr Ballard became the first person ever to solo climb all six major north faces of the Alps in one winter.
He had been living in Italy’s Dolomites mountain range with his father for the last few years.
Friend of the family Chris Terrill told the BBC they were a “mountain family” and said he accompanied Jim Ballard and his children on a trip to K2 after Ms Hargreaves died.
“It was an extraordinary expedition and it ignited something in Tom,” he said.
“And no-one was going to stop him from following in his mother’s footsteps.
“As tragic as his death is, he died doing what he loved.”
One of Britain’s most experienced climbers, Alan Hinkes, who knew Mr Ballard’s mother, described their deaths as a great loss.
“This is one of the most dangerous, difficult mountains in the world, and in winter, I think if anything goes wrong, it happens pretty quickly,” he added.