Celtic's official Twitter account said: "Everyone at Celtic is deeply saddened to hear of the passing of former midfielder Liam Miller".
Sir Alex, who signed the Republic of Ireland midfielder for United in 2004, said: "I am very sad to hear that Liam has died. So very sad. Thoughts and prayers with his friends and family".
A-League clubs Glory, Roar and City have also paid tribute to the midfielder following a four-year stint in Australia during the latter stages of his career.
Miller also won 21 caps for the Republic of Ireland at global level.More news: Indian PM arrives in Ramallah for talks with Abbas
"On behalf of all my staff who worked with Liam at the time, we send our deepest sympathy to his family".
Celtic said they were "deeply saddened" by the news and will hold a minute's silence before this weekend's game.
Former England captain and teammate of Miller's at Man Utd, David Beckham, said his thoughts were with Miller's family, adding: "Rest In Peace". "So sad. Thoughts with his family".
His former team-mate at Cork City, Alan Bennett, also posted an emotional message.
"At the end of that season, Liam moved on to Manchester United but had already made a strong mark at Celtic Park and the players were extremely disappointed to see him go".More news: Policeman killed, two wounded in Georgia shooting
Ex-international teammate Kevin Kilbane said, "Sad hearing the bad news Liam has passed away, taken far too soon by a terrible disease".
He went on to play 60 times for Sunderland, scoring three times and invariably being a quality player, who while quiet and reserved in public, was always enormously popular in the dressing room.
Fine Gael councillor Des Cahill told Independent.ie: "As a fan I particularly remember him playing for Celtic and latterly for Cork City". His young children & wife, his family, his friends lives changed forever.
"The news, we heard about it just recently, we didn't know how bad it was".More news: 'Black Panther' London premiere: All the red carpet looks