Chirivella, who signed for Liverpool academy at the age of 16 from Valencia, has made five first-team appearances for the side.
The midfielder was given his debut in the Premier League by Klopp at the end of the 2015/16 season.
He was then send out on loan to Dutch side Go Ahead Eagles.
Chirivella impressed at the Netherland's side last season.
He made 31 appearances in the Eredivisie during a loan move to Willem II, where he helped the side retain top-flight status.
The 21-year-old was part of the Liverpool pre-season team, and came on against Manchester United in their 4-1 victory.
Rosenborg, who is managed by former Liverpool favourite Stig Inge Bjornebye, host Celtic in the Champions League second-leg qualifier on Wednesday.