Parents Coral and Paul Jones said police officers warned them of an initial hearing at the Court of Appeal next month, which the couple plan to attend.
It is not known if the appeal is against his sentence or conviction.
Mrs Jones said her family continue to be "tortured" by Bridger, who was jailed for life in May after being found guilty of the girl's murder.
"It's disgusting," she said. "He's in prison where he belongs and he should stay there. He's just torturing my family with these legal battles.."
She added that Bridger had already tried to claim compensation following an attack by a fellow inmate while at the top security Wakefield Prison, West Yorkshire, earlier this year.
Bridger, a father of six, snatched April near her home in Machynlleth, Wales, on October 1 last year.
The five-year-old's body has never been found.