“We are supporting each other the best we can as a family and ask for privacy and space at this awful time,” they said.
Liam was one of five members of One Direction.
The pop star, who found fame on The X Factor in 2010, died after falling from the third floor of a hotel in Argentina, police say.
Liam was one of five members of One Direction, along with Zayn Malik, Louis Tomlinson, Niall Horan and Harry Styles.
Following his death at around 17:00 (21:00 BST) on Wednesday, stars expressed their upset and posted memories of the dad who shared a son, Bear, with former partner Girls Aloud star Cheryl.
Former bandmate Harry’s mother Anne Twist paid tribute, posting a broken-hearted emoji on Instagram, captioning it: “Just a boy.”
Singer Rita Ora, who collaborated on a song with Payne in 2018, said she was “devastated” in a post on Instagram, adding that she “loved working with him so much”.
On Thursday, police in Buenos Aires said a preliminary autopsy suggested the One Direction star died from external and internal bleeding injuries.
They said the area where Payne fell had been inspected and items including alcohol and a phone had been discovered. Medication was found in his room.
Olly Murs, who starred on The X Factor a year before One Direction, also shared his condolences on Instagram, saying he was “lost for words” and described Payne’s death as “devastating”.
Murs said they “always had a good laugh” when they met, mostly talking about “how annoyingly good his hair always looked, or our love for Becks, the old XF [X Factor] days and the tour we shared together.
“Liam shared the same passions as me, the same dreams, so to see his life now end so young hits hard. I’m truly gutted and devastated for his family and of course his son Bear losing a dad.”
Dermot O’Leary, who hosted The X Factor when Liam appeared, posted a photograph of the pair on stage, captioning the Instagram post: “The worst news.”
“I remember him as a 14-year-old turning up to audition on The X Factor, and blowing us away singing Sinatra. He just loved to sing,” he wrote.
“He was always a joy, had time for everyone, polite, grateful, and was always humble.”
