Known as ‘Mr Football’, Murray has been largely credited with making the World Game part of Australian culture.
He leaves behind two daughters, Tania and Natalie.
Opposition Leader Bill Shorten said the family should be offered a state funeral.
"Les is a national icon who did so much to grow the world game and to champion multicultural Australia. He deserves nothing less than the full symphony of tribute and respect," Mr Shorten said in a statement.
"Ultimately it’s a decision for Les’s family, but I think the offer should be made at least. I can think of few people as deserving of this high honour."
Murray began at SBS in 1980 as a Hungarian subtitle writer.
But his passion for football rapidly brought him into the sports team. By 1986, he was hosting the network's World Cup coverage.
As an SBS football commentator and presenter, he covered eight World Cups, before retiring in 2014. That same year he was also inducted into the FFA Football Hall of Fame.
Tributes have flown from all circles for the football commentator.
Prime Minister Malcolm Turnbull issued his condolences in a Facebook post, saying the Socceroos and Matildas never had a more devoted fan than Murray.
"You championed the beautiful game - The World Game - in Australia over so many years," Mr Turnbull said.
"You and Johnny Warren really made The World Game Australia's game. Rest in peace."
SBS Managing Director Michael Ebeid said it was a sad day not just for the multicultural broadcaster, but sports broadcasting and the football community at large.
“Les was certainly an iconic football presenter, and he did an amazing job for us over 37 years," Mr Ebeid said, paying tribute to Murray's passion to nurture and develop up-and-coming writers and broadcasters.
"Les has mentored many young journalists and young commentators in football over the years," Mr Ebeid said.
"And the one thing he often used to say is 'it's not about us hosts, it's not about us in entertainment, it's always about the game'."
Football 'won't be the same'
Football Federation Australia said few people besides Murray became synonymous with their sport.
"He literally brought the game to millions and connected us to the biggest sport in the world," FFA chairman Steven Lowy said.
“I’ve been seeing his face and hearing his voice since I was a teenager and I am sure I speak for many when I say that football is not going to be quite the same without him around."
Friend and former SBS colleague Andy Paschalidis said Murray left a lasting legacy for all football fans.
"He spread the message of the World Game better than anyone before him during his time, and dare I say beyond his time."
Former refugee and diversity advocate
Murray emigrated to Australia in 1956 as an 11-year-old, as his family escaped Hungary amidst communist Russia’s occupation. In 2015 he told SBS that people smugglers helped them escape.
The family first landed in Darwin, before eventually arriving at a migrant camp near Wagga Wagga.
Former colleague and close friend George Donikian said Murray was dedicated to promoting diversity in Australia.
"The work that Les did to support multicultural Australia was fantastic. Immense in fact," Mr Donikian said.
"His name and his legacy will be that he actually transcended SBS and transcended so many different platforms."
The Australian Multicultural Foundation said Murray was "a great advocate for multiculturalism" and "an inspiration and mentor to millions of young people in Australia".
Sports commentator Ned Hall said Murray was widely respected in sport broadcasting circles - doing much to bring mainstream attention and transparency to the game.
"I think it just goes to show the respect and esteem in football in Australia and also globally as well that he rose to that position within FIFA," Hall told the ABC.
"And I say within FIFA reservedly because he was a big critic of the corruption and the way it had gone in more recent years as well."
"So he was quite willing to speak out the failure of processes that impacted the World Cup bids and particularly in recent times.
"But I think it was all out of that love for the game, the passion for game, globally and in Australia as well he just wanted the best of football."