Koltsov, a former ice hockey player who competed in the NHL for the Pittsburgh Penguins, had been a regular presence supporting Sabalenka at tournaments.
The world No.2 frequently referenced Koltsov on social media in pictures and messages.
The news was announced by Russia ice hockey team Salavat Yulaev Ufa, where Koltsov had been assistant coach.
A statement on the club's website read: "It is with deep sorrow that we inform you that Salavat Yulaev coach Konstantin Koltsov has passed away. He was a strong and cheerful person, he was loved and respected by players, colleagues, and fans.
"Konstantin Evgenievich forever wrote himself into the history of our club. Koltsov won the Russian Championship and the Gagarin Cup as part of Salavat Yulaev and did a great job on the team's coaching staff.
"The hockey club Salavat Yulaev expresses its condolences to the family and friends of Konstantin Evgenievich Koltsov."
Koltsov represented Belarus at the 2002 and 2010 Winter Olympics and was also a coach of the national team.
A statement on the federation's website said: "The Belarusian Hockey Federation expresses its deepest condolences to the family, friends and everyone who knew and worked with Konstantin Evgenievich."
Pictures showed tributes to Koltsov outside Ufa Arena laid by fans of Salavat Yulaev.
Sabalenka, who won her second grand slam title at the Australian Open in January, is in Miami and is due to play her first match at the Miami Open later this week, although her participation must now be in severe doubt.
It is the second tragedy to hit the 25-year-old Belarusian, whose father Sergey, also a former ice hockey player, died in 2019 at the age of 43.
Sabalenka's victory at the Australian Open fulfilled the dream she shared with her father of winning two slam titles by the age of 25 and, speaking in Melbourne, she said: "It was really important. Of course he's my biggest motivation. He's been everything for me."
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