The pair were among eight players to attend the annual scholars night on Friday, which saw youngsters inducted into the Professional Development Phase group.
Carrick and Rooney posed for a photograph together on the night, while their proud fathers watched on.
Manchester United manager Michael Carrick was there both as first-team boss and as father to Jacey, while Wayne Rooney was present on a night off from World Cup punditry duty.
Kai, 16, made six appearances in the Under-18 Premier League last season as well as making his FA Youth Cup debut.
He is expected to play a more prominent role for Darren Fletcher's Under-18 side next season.
Jacey, also 16, made just one appearance in the Under-18 Premier League last term, but the midfielder has signed scholarship forms to continue his development next term.
A scholarship contract allows players to continue their development as footballers and is the final step before the possibility of signing professional terms, which can come into effect from a player's 17th birthday.
Often, young players will agree professional contracts before they turn 17, with those deals beginning from their birthday.
The other players to sign scholarship deals this summer are Gazik Ibragimov, Edson Dejonge-Seiros, Harlem McLaughlin, Pharell Silvester, Connor Laurie and Jaume Camacho.
At 15, JJ Gabriel is too young to sign, and that development will come next season, when United could again face a fight to keep their talented attacker.
The scholars night is an annual tradition at Old Trafford, giving young players and their families the chance to mark the milestone of officially joining the club's Professional Development Phase.
Both Carrick and Rooney carry famous surnames, with their fathers each having made hundreds of appearances for United's first team during their playing careers.
United's academy continues to be a key part of the club's long-term planning, with the eight scholars now set to begin the next stage of their development together at Carrington over the coming season.