Rugby, not just hockey, has been in Burke's heart for decades. He's played it, watched it and will now take part in directing it as well.
Rugby Canada announced Tuesday that Burke is joining its board of directors, giving the former NHL executive a chance to get involved in what he calls his "second favourite sport."
"I'm really excited about it," said Burke. "I think it's a great sport and I think the future of rugby in Canada is very bright."
The former general manager of the Toronto Maple Leafs first played rugby following a brief playing career in hockey. When he was a 23-year-old law student, Burke said he noticed the Harvard Business School Rugby Football Club practising near where he lived and asked if he could join.
"I never played before, never watched before. My older brother played a little bit but I wasn't familiar with the sport," he said. "But I learned to love it. I played for two years after I graduated even though I was practising law."
Now 58 years old, Burke is an avid fan of the game.
He said he goes to as many of Canada's international games in Toronto as he can, and once even saw the senior men's team play Japan in Bordeaux, France in 2007.
"Brian has a deep understanding of professional sports in North America and, with his successes and contacts across the National Hockey League and beyond, we're happy to have him involved with Rugby Canada," Graham Brown, chief executive officer of Rugby Canada, said in a release.
Burke has had a long relationship with Rugby Canada. He's spoken at several events put on by the organization and said he's known former president Rick Bourne since around 1990.
He said his first priority will be to get more familiar with the senior women's team, who captured its first Nations Cup title earlier this month after beating England, 27-13, for the first time in 20 years.
Burke is already impressed with the men, who defeated the U.S. Eagles 27-9 on Sunday in the first leg of qualifying series for the 2015 Rugby World Cup in England.
Canada, ranked 15th in the world, hosts the U.S. in the finale Saturday at Toronto's BMO Field. The winner of the total-points series advances directly to the World Cup, while the loser gets another shot with a series against Uruguay.
"They're beating sides that used to beat them easily, they're in most matches. They're not in the elite group yet but they're on their way," he said.
Burke also sees potential in rugby sevens, which will make its Olympic debut at the 2016 Games in Rio de Janeiro.
"It bears very little resemblance to 15s. But I think to me, the most exciting thing about sevens is we can attract an audience with our sevens program," he said. "We can get people watching rugby to say, 'This is cool,' and then next thing you know they're watching 15s."
Prior to his tenure with the Leafs, Burke served as a GM with the Hartford Whalers, Vancouver Canucks and Anaheim, winning a Stanley Cup with the Ducks in 2007.
Hockey is still the priority for Burke, who is currently a part-time scout with the Ducks. A dual citizen, Burke was also GM for the United States men's team that won a silver medal at the 2010 Winter Games and is the director of player personnel for the 2014 Sochi Games.
Follow Tyler Harper on Twitter at @tyler_harper