Dr Choje Akong Rinpoche was stabbed along with his nephew and their driver in Chengdu, following what Chinese police are saying was a dispute over money.
He was the abbot and director of the Samye Ling Monastery in Eskdalemuir, Dumfries and Galloway, Europe's biggest and oldest Tibetan religious centre.
Police in Chengdu say the killings took place in a residential area and three men, all Tibetans, have been arrested in connection with the deaths.
A statement posted by his brother, Lama Yeshe Rinpoche, on the monastery's website said:
"Rinpoche's body has been taken to hospital where a post-mortem will be carried out.
"We will have to do a lot of special prayers and make a lot of appropriate offerings on Rinpoche's behalf and any contributions you wish to make in his name will be much appreciated. I request you all to do whatever prayers you can."
A statement from the police in Chengdu said: "The three suspects have confessed to the crime and the case is still under investigation."
The Dalai Lama, who gave his blessing to Samye Ling when it was founded, said he would offer prayers following Dr Rinpoche's death.
The Buddhist leader the Karmapa, a friend of Dr Rinpoche, wrote on his website: "Akong has been my friend from the time I was seven. A social activist, he showed great kindness to Tibet by founding schools and hospitals, printing old texts, and helping many people.
"Thus I am shocked to hear that he along with two others has been taken from us so suddenly. I would like to offer my condolences to his family members, everyone at Samye Ling Monastery, the Rokpa Foundation, and all the persons involved in his projects in Tibet as well as to all of the students whose lives he touched.
"I hope that all of his visions and aspirations may continue to be fulfilled."
Dr Rinpoche founded the Samye Ling Monastery in 1967 alongside his brother and Sherab Palden Beru, who died last year aged 101.
The main temple, which holds 500, was officially opened by the then Borders MP David Steel, now Lord Steel of Aikwood, in 1988.
As well as owning the large main retreat in Eskdalemuir, the centre also owns Holy Island off Arran and retreats in London, Dublin and Brussels and has attracted support from celebrities including Richard Gere, Billy Connolly, Koo Stark and David Bowie,
Dr Rinpoche attended the 60th anniversary of the signing of the UN Refugee Convention with Home Secretary Theresa May in London in 2011.
He said that he was welcomed by the British Government and its people when he came as a refugee in 1963 and he presented Mrs May with a long white scarf endorsed with messages of friendship.
Dr Rinpoche also acted as a consultant for the National Museum of Scotland when its collection of Tibetan exhibits was being relocated during its recent £46.4 million refurbishment.
The Samye Ling Monastery attracts thousands of visitors every year, many of whom stay for weekend breaks in public accommodation.
About 60 people, including monks and volunteers, are thought to live in it full-time.
A spokesman for the Foreign Office said last night: "We can confirm the death of a British national in Chengdu, China, and we stand ready to provide consular assistance.
"We are not aware of any further details at the moment."