The defender, who won the Europa League last season while on loan at Sevilla, will be reunited with former United team-mate Cristiano Ronaldo at his new club.
Financial details were not disclosed although media reports have suggested that Man Utd would receive around £6 million for the player.
"A new adventure beckons for @AT13Oficial. Thank you for everything and good luck, Alex," United posted on Twitter.
Al-Nassr released a photograph of the defender holding their yellow jersey and a video in which Ronaldo could be seen smiling in the background.
The Saudi Pro League has attracted many players since Ronaldo joined Al-Nassr last December in a mega deal.
Al-Nassr, who finished second in the league last season, were banned by FIFA earlier this month from registering new players due to outstanding debts.
According to reports, the outstanding debts were related to the transfer of Nigerian attacker Ahmed Musa, who moved to Al-Nassr from English side Leicester City in 2018.
In a ruling in October 2021, FIFA said Al-Nassr still had to pay a sum of €460,000 euros to Leicester for the signing of Musa, who played 62 games for the Saudi Arabian club in all competitions, scoring 14 goals, before leaving in 2020.
