Strictly judge Craig Revel Horwood added: “Extremely sad to hear the news of Bruce’s passing. A true legend and national treasure. He will be deeply missed but always remembered.”
Former judge Arlene Phillips said the entertainer was an “indestructible Titan, tap dancing his way through life”.
She added: “Working with him on Strictly was personally a joy. His endless teasing of my judging style, particularly with strong sportsmen, was an ongoing joke.
“His enormous support after I was let go from the Strictly panel meant so much.”
In a tweet, Prime Minister Theresa May said the country had “lost a national treasure”, adding: “Like millions of others, for years I watched Sir Bruce dance, sing, joke & laugh. He will be sorely missed.”
BBC director-general Lord Hall described Sir Bruce as “one of the greatest entertainers our country has ever known”.
Comedian Jimmy Tarbuck added: “He could do it all. He was magnificent and he was a great entertainer. He could dance, he was a very nice pianist, he was good at sketches, he was the greatest moaner in the world on the golf course, and he was a unique friend to me.”