Linee, who was suffering from Motor Neuron Disease (MND), was diagnosed in 2013.
He played 112 matches at centre for his province between 1992 and 2001 and made his Springbok debut in 1993, at the age of 23.
He played nine tour matches for his country in Australia, Wales, Scotland, and Ireland between 1993 and 1994, but was never capped in a Test.
He will be remembered as a loyal Western Province player with a no-nonsense approach to defence as displayed by his bone-crunching tackles, the union said on its website.
Sport Minister Fikile Mbalula said he was saddened to hear of Linee’s death in the early hours of the morning.
“We wish to extend our deepest condolences to the Linee family as well as the rugby and sporting fraternity,” he said in a statement.
“We shall be interacting with Saru and the family for further information relating to his memorial and burial.”
Western Province Rugby Football Union president Thelo Wakefield told www.wprugby.com: “This is a sad, sad day for Western Province and South African Rugby.
“Tinus fought a very brave battle against a dreadful disease and his tenacity and guts will always be remembered, as a rugby player and during the past 18 months.
“Our thoughts are with his entire family, especially his wife Diana, who stood like a rock beside her husband.”
Joost van der Westhuizen, former Springbok scrumhalf and fellow MND sufferer expressed his condolences on social media.
“Sad day for Rugby in SA. Just received news of the passing of a rugby legend and MND Warrior Tinus Linee RIP my friend #J9Foundation,” he said on Twitter.
After hanging up his boots, Linee worked as a development officer for the union and was a member of the WP Vodacom Cup management team in 2008.
Linee is survived by his wife Diana, her two sons, and a daughter from his first marriage.