Laporte played 12 Premier League games for Manchester City last season
The 29-year-old joined City from Athletic Bilbao in January 2018 for a then club record £57m.
He helped City win 12 major honours but made just 24 appearances as the Blues won the Treble last season.
In a goodbye message on social media, Laporte thanked fans and team-mates for an "unforgettable" time at the club.
"It has lasted five-and-a-half unforgettable years. Plenty of memories that I will keep forever close to my heart," said Laporte.
"From trophies to setbacks, victories to injuries, it's all been part of my journey, and I'm grateful for absolutely every moment that shaped me into who I am today.
"It was an honour and a pleasure to wear these colours and I hope you will remember me in a good way. I will always be a City fan and I look forward to seeing you all again."
Laporte made 180 appearances for City in total, scoring 12 goals, but he found opportunities limited last season.
Coach Pep Guardiola opted to use defenders Manuel Akanji and Nathan Ake more often, and City spent £77m to bring in Croatia defender Josko Gvardiol earlier this month.
"Aymeric has been an excellent player for Manchester City during a great time for the club," said City's director of football Txiki Begiristain.
"He fit into the team seamlessly upon arrival in 2018 and played a key role in many of our successes over the years that followed.
"The form of the other defenders in the squad made it hard for him to find sufficient game time in 2022-23 and we respect his decision to seek a move elsewhere."
Laporte switched his international allegiance from France to Spain in May 2021 and already has 22 caps, helping them win the Uefa Nations League final in June.
He is the latest high profile player to move to Al Nassr, joining Cristiano Ronaldo, Sadio Mane, Marcelo Brozovic, Alex Telles and Seko Fofana at the club.
They are one of four Saudi Pro League clubs to be taken over by Saudi Arabia's Public Investment Fund (PIF) in June, the others being Al-Ittihad, Al-Hilal and Al-Ahli.
Last month, Laporte's former City team-mate Riyad Mahrez joined Al-Ahli in a deal worth up to £30m, while Karim Benzema, N'Golo Kante, Ruben Neves and Neymar are among other footballers to move to Saudi Arabia this summer.
-- Courtesy of BBC Sport