Three suspects, all ethnic Tibetans, have been detained by police, Hong Kong-based South China Morning Post quoted a police statement as saying.
Police said a monetary dispute caused the altercation.
"Following a verbal dispute between the two sides, the three suspects stabbed the three victims to death with knives they were carrying," the statement said.
In a statement, the Samye Ling monastery confirmed the report saying its founder has been "assassinated".
The statement was signed by the monk`s brother Lama Yeshe Rinpoche.
"Rinpoche`s body has been taken to hospital where a post-mortem will be carried out," Lama said.
Born in the Chamdo prefecture in Tibet in 1940, Rinpoche was identified at a young age as the reincarnation of the abbot of the Drolma Lhakang monastery, where he was later raised.
After the 1959 uprising, he fled to India. Two decades later, he co-founded the Kagyu Samye Ling Monastery in Dumfries and Galloway near the English-Scottish border.
The Karmapa Lama, the second most important figure in Tibetan Buddhism, expressed his condolences over his death in a statement, the Post report said.
Rinpoche had led the search which identified the Karmapa at the age of seven as a reincarnation of the previous Karmapa. The Karmapa fled Tibet in December 1999 and has since lived in India.
Tibet region experienced tensions in the recent months over spate of self immolations which the Tibetan overseas groups say was aimed at protesting against the Chinese rule.
Over 100 self immolations have taken place in different parts of Tibet.
A number of monks and locals have been arrested for instigating the immolation protests.