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Ablett, who had a season as Stockport manager in 2009-10, lost his 16-month fight against non-Hodgkin's lymphoma on Sunday night.
"On behalf of Gary Ablett's family the LMA has today confirmed that Gary passed away peacefully last night following a hard fought 16-month battle against non-Hodgkin's lymphoma," said an LMA statement.
"Respected throughout the game by players, coaches and managers alike Gary will be sadly missed.
"The LMA's thoughts and condolences are with Gary's wife Jacqueline and children."
Liverpool manager Kenny Dalglish, who gave Ablett his debut during his first spell as Reds boss, offered his condolences.
"It's a sad, sad day for his family and everyone connected with Liverpool Football Club," he told liverpoolfc.tv. "The most important thing just now is to pay our respects to Gary, his wife Jacqueline, the two boys and his wee girl.
"Obviously he had a long battle and I'm sure it was a lot of suffering and a lot of pain for him but the only thing you can say is at least he won't be suffering any longer.
"I gave Gary his debut and remember him scoring on his first start at Anfield against Nottingham Forest.
"He was a really good servant to the football club not only as a player, but also as reserve team coach.
"He served the club proudly and credibly. It's very sad for everybody."
Former Liverpool manager Roy Evans told Sky Sports News: "It's a big shock. Obviously we know Gary's been ill for a while. It's a really sad day."
Recalling Ablett as a player, Evans added: "He was good at everything he did as a player.
"He went into coaching and stamped his personality onto that and was doing a very good job, certainly at youth level. I remember he had bit of a difficult first spell at Stockport, but overall he's had a great career in football, he's a big loss to the sport.
"He made a big contribution to the game."
David Fairclough, who played for Liverpool between 1974 and 1983, added: "He was a lovely lad, he always conducted himself in the right way. He was great with people, always personable. What always came through with Gary was his attitude was just always very, very right."
Ablett took up a coaching role with Ipswich in July 2010 but was taken ill while at the club's training ground and was diagnosed with non-Hodgkin's lymphoma.
"Liverpool Football Club were today saddened to learn of the death of former player Gary Ablett," said a statement from the Reds. "The staff of Liverpool FC would like to send our condolences to the family and friends of Gary."
A statement from Everton said: "Everyone at Everton was saddened to learn of the death of Gary Ablett.
"The thoughts of everyone at Everton are with Gary's family and friends at this tragic time."
In October Ablett spoke of the shock at being diagnosed with the disease.
"Jacqueline took one look at me and took me straight to the doctors," he said in an interview with Everton's official website to support Blue September - a campaign to raise awareness of forms of cancer specifically affecting men.
"The following day they whisked me to Ipswich hospital and the sister asked us if we knew why we were there
"We said 'for tests' but she told us they had found a really aggressive lymphoma and that without immediate treatment I'd be in serious trouble.
"From diagnosis to that first session of chemotherapy was less than three days, so it was a massive shock."
Former Liverpool and Germany midfielder Dietmar Hamann, who became Stockport manager in 2011 a year after Ablett left the post, said on Sky Sports News: "It's devastating news. He's a very highly regarded and respected man on Merseyside and a true legend."
Paul Walsh, who played alongside Ablett at Liverpool in the 1980s, described the news of Ablett's death as "absolutely tragic".
He said: ""I'm absolutely stunned at the news, I knew he wasn't well, but 46 years of age is absolutely tragic."
"He was was a hell of a nice person. He was a top lad. Just wanted to get in there and do well, didn't do anything fancy but a great pro who trained hard with a great attitude to everything he did.
"There was also a steely determination about Gary. You don't get in the Liverpool team and come through the ranks at Liverpool and play 100 times for them if you haven't got that little bit of steel."
Michael Owen said on his Twitter account: "So sad to hear the news that Gary Ablett has passed away today aged 46. He played for some great clubs and had a career to be proud of. RIP."
Former club Birmingham have announced a minute's silence will be observed before their Championship match at Peterborough today with another held at St Andrew's on Saturday prior to their FA Cup third round tie with Wolves.
"The thoughts of everyone connected with Birmingham City Football Club are with the friends and family of Gary Ablett," said a Blues statement.
"Just like he battled as a fierce competitor on the pitch, Ablett fought non-Hodgkin's lymphoma for over year before passing away peacefully last night.
"Our deepest sympathies and condolences go out to Gary's wife Jacqueline and his children and everyone else that knew him."
Ipswich's players will wear black armbands for this afternoon's visit of Nottingham Forest and Ablett will be honoured with a minute's applause.
"It's desperately sad news," Town chief executive Simon Clegg told itfc.co.uk.
"I stayed in touch with Gary as he battled his illness over 18 months, which he did with bravery and dignity and our thoughts go out to his wife Jacqueline and their children at this very difficult time."
Ian Rush, who played alongside Ablett during his two spells at Liverpool in the 1980s and 90s, said Merseyside will be united in their sorrow today.
"He sums up what people are like in Liverpool, In 1989 (FA Cup final) when Liverpool and Everton fans were together, this will be exactly the same for Gary," Rush said.
"He's a true Scouser and when things are going wrong people stick together, they get together, both Liverpool and Everton supporters will support each other. That's what people on Merseyside will be doing today."
He added on Sky Sports News: "We're absolutely gutted, Gary was a gentleman both on and off the pitch. When I received the news this morning I just couldn't believe it. It was terrible news and my heart goes out to his family.
"What you saw is what you got with Gary. Whoever he played for, be it Liverpool, Everton, Birmingham or whoever, he always gave 100%. That's very rare these days. In those days people just saw him as a normal player, but I think in today's game he'd be right up there because these sorts of players don't come by anymore."
Rush also praised Ablett's coaching ability, adding: "People don't realise what a good coach he was.
"I remember him when he was managing Liverpool reserves, he put his voice across in the right way. He wasn't a shouter or a raver, he put it in the right way. His knowledge of the game was getting better all the time.