Mani's brother, Greg, wrote in a post on Facebook: "It is with the heaviest of hearts that I have to announce the sad passing of my brother."
Former bandmate Ian Brown posted "rest in peace Mani" alongside a kiss, while Tim Burgess, the lead singer of fellow indie band The Charlatans, said the late musician "never failed to bring a smile to my face".
Mounfield was part of The Stone Roses' classic line-up alongside singer Brown, guitarist John Squire and drummer Alan 'Reni' Wren.
He worked on the Manchester band's well-known 1989 self-titled debut album, which featured hits such as I Wanna Be Adored, She Bangs The Drums and I Am The Resurrection, which is underpinned by his prowling bassline, and its harder-rocking 1994 follow-up Second Coming.
Liam Gallagher, the lead singer of Oasis, who have often quoted The Stone Roses as an influence, said on X that he was in "total shock" about Mounfield's death.
He said he was "absolutely devastated", describing the bass player as "my hero".
After The Stone Roses broke-up in 1996, citing musical differences, Mounfield joined Scottish rock band Primal Scream, first playing on their album Vanishing Point, released a year later.
Mounfield would go on to record four more albums with the band, including their electro-punk classic XTRMNTR in 2000, Evil Heat (2002), Riot City Blues (2006), and Beautiful Future (2008), before leaving the band in 2011 to reform The Stone Roses.
Following news of his death, Primal Scream posted a picture of him on their Instagram page, while The Stone Roses posted a joint tribute on their account.
Echo And The Bunnymen singer Ian McCulloch also posted a tribute to the bass player on X, saying his "thoughts and feelings" went out to his family.
McCulloch said: "I'm absolutely gutted to hear the news about Mani, who I have always loved and always will love, deeply and forever. Like a brother.
"I am in shock to be honest. Please tell me I'm just having a bad, bad dream."
Shaun Ryder, the lead singer of Happy Mondays, who were contemporaries of The Stone Roses, said in a post on X: "RIP Mani - my heartfelt condolences to his twin boys and all of his family."
With the reformed band, the bass player went on a reunion tour around the world that year, which ran until 2013, and later recorded the comeback singles All For One and Beautiful Thing in 2016.
The band played a number of UK gigs in 2016 and 2017, including concerts at Manchester's Heaton Park, Wembley Stadium and Glasgow's Hampden Park, which would end up being the classic line-up's last concert with Brown telling the crowd: "Don't be sad that it's over, be happy that it happened."
Mounfield married his late wife Imelda after the two met during the 13-month recording of Second Coming at Rockfield Studios in Monmouth, Wales, and they had two sons together.
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