An AFP correspondent saw the Netherlands international enter the club's headquarters in the eastern city of Al Khobar on Tuesday.
Reports said Al Qadsiah paid around 61 million euros (£52.2m) for the 28-year-old, who represented the Netherlands at the 2026 World Cup in North America.
Reijnders had been under contract with City until June 2030 after joining the Premier League club from AC Milan last year. He trained separately from the first-team squad last week before missing City's 3-0 defeat by Arsenal in the FA Community Shield.
His departure is the latest change in a significant midfield reshuffle at City following Rodri's move to Barcelona. The English club have also been linked with Chelsea and Argentina midfielder Enzo Fernandez.
Reijnders made 47 appearances in all competitions under former City manager Pep Guardiola last season, scoring seven goals.
Al Qadsiah are backed by Saudi state-owned oil company Aramco and coached by former Liverpool boss Brendan Rodgers.