The 72-year-old died in the early hours after a short illness, his son Paul Martin, a former MSP, confirmed.
Mr Martin said: “He passed away early this morning after a period of short illness. He passed away peacefully with his family.”
Labour Leader Jeremy Corbyn offered his condolences, saying he was “very sorry” to hear the news.
John Bercow, the current House of Commons Speaker, said: “I’m very sorry to hear this news.
“Michael was a decent, public-spirited man who had a real care and concern for members, their staff and the staff of the house.
“He was a good campaigner and both passionate about and proud of his roots. He also had a great sense of humour.
“On a personal level, he was always very kind to me. I still remember the lovely letter of congratulations he sent to me after my election as Speaker.
“My deepest sympathies go out to Mary, his children and grandchildren.”
Disparagingly dubbed “Gorbals Mick” by some because of his Glaswegian accent, he was described as an “authentic working class” voice in the House of Commons by current Scottish Labour leader Richard Leonard.
Former prime minister Gordon Brown said Lord Martin will be “sorely missed”, adding: “I knew him as a successful young trade union organiser, a dedicated member of Parliament, a conscientious Speaker of the House of Commons and latterly as a member of the House of Lords and no matter the position he occupied, he never forgot the debt he owed to his upbringing and the people he had been elected to serve.”
Michael Martin – latterly Lord Martin of Springburn, after the Glasgow constituency he represented from 1979 to 2005 – was Speaker from 2000 to 2009.
In October that year, the former Labour MP joined his predecessor as Speaker, Baroness Boothroyd, on the crossbenches. Although by tradition the Speaker is elevated to the Lords, the decision to grant Lord Martin’s peerage proved controversial, with some speculation the honour should be withheld.
When he was Speaker, he was criticised for leading efforts to block the publication of MPs’ expenses, and for his handling of the affair once the revelations began.
He was the first Speaker to be forced to resign – over the expenses issue – in 300 years.