Cilenis Marulanda was freed almost immediately after being taken by guerrilla group the National Liberation Army earlier this month but father Luis Manuel was held captive for 12 days until being released late last week.
Diaz was reunited with his parents when he returned to Colombia for international duty and the pair were in the crowd to see their son score both goals in an emotional 2-1 victory at home to Brazil on Thursday.
The PA news agency understands that following the game Liverpool paid for a private jet, for security reasons and to help manage the familys trauma, to bring Diazs parents and wider family to Merseyside so they could spend an extended period of time together.
Their plane was scheduled to land at Liverpools John Lennon Airport early on Saturday morning.
Since Diazs parents were kidnapped, the club have endeavoured to provide the best possible support, with manager Jurgen Klopp allowing Diaz compassionate leave and leaving decisions about when he trained and was available for selection entirely up to the player, in keeping with the family values they believe are vital to the clubs ethos.
Diaz will rejoin his family in Liverpool when he returns from international duty after their match in Paraguay on Tuesday.
He will be on a flight with the clubs other South American players Alisson Becker, who was beaten twice by his club-mate in their game in Barranquilla in the far north of Colombia, Alexis Mac Allister and Darwin Nunez who faced each other in Argentinas 2-1 defeat by Uruguay in a plane jointly chartered by a number of Premier League clubs to return their players home as expediently as possible.
Liverpool head to Manchester City in a top-of-the-table clash at the Etihad Stadium next Saturday lunchtime.