Rangers, with whom the former Dundee United defender had two spells as manager from 1991-98 and 2007-11, offered their condolences to his family in a statement on the club's official website.
"It is almost impossible to encapsulate what Walter meant to every one of us at Rangers," club chairman Douglas Park said. "He embodied everything that a Ranger should be.
"His character and leadership was second to none, and will live long in the memory of everyone he worked with during his two terms as first team manager."
Walter Smith— Rangers Football Club (@RangersFC) October 26, 2021
Smith won 10 league titles with Rangers, including seven in a row, as well as five Scottish Cups and six League Cups. He also took Rangers to the 2008 UEFA Cup final.
He remains the club's second most successful manager after Bill Struth.
Smith was assistant to Sir Alex Ferguson, then Scotland manager, at the 1986 World Cup in Mexico and became manager of the national team in 2004, following on from German Berti Vogts.
He returned to Rangers in 2007 after Scotland failed to qualify for the 2006 World Cup finals.
The Scottish FA released a statement on Tuesday which read: "We are deeply saddened to learn of the passing of the former Scotland national coach, and one of the most successful managers in Scottish football history, Walter Smith.
We are deeply saddened to learn of the passing of the former Scotland national coach, and one of the most successful managers in Scottish football history, Walter Smith.— Scottish FA (@ScottishFA) October 26, 2021
The thoughts of everyone at the Scottish FA are with his friends and family at this time. pic.twitter.com/CP2TzXLm51
"The thoughts of everyone at the Scottish FA are with his friends and family at this time."
Smith's four year stint in the Premier League with Everton ended with his dismissal in March 2002. The club also paid tribute to Smith on social media on Tuesday.
Smith also spent time as Manchester United assistant coach in 2004 and the club released a statement which read: "Everyone at Manchester United is deeply saddened to hear of the passing of Walter Smith.
"Our thoughts and prayers are with Walter's friends and family at this extremely difficult time."