Veiga, a product of Celta's academy, scored 11 league goals for the Spanish side last season, including a brace on the final day of the campaign in a 2-1 win over champions Barcelona - which helped his side avoid relegation.
Veiga, who has played eight times for Spain's Under-21 team and was part of the squad which finished runners-up at the U21 European Championship last month, has attracted interest from several clubs in the transfer window including Chelsea, Liverpool and Napoli.
"The jewel of Spain will chant our anthem," Al Ahli posted on messaging platform X, previously known as Twitter.
Veiga is the latest addition to Al Ahli's squad after the signings of Riyad Mahrez (32), Roberto Firmino (31), goalkeeper Edouard Mendy (31), Alain Saint-Maximin (26), Merih Demiral (25), Franck Kessie (26) and Roger Ibanez (24).
Al Ahli, who have returned to the Pro league following a season in the second division, have enjoyed a great start to the season with three wins from three games.
The Saudi Public Investment Fund (PIF) has announced a Sports Clubs Investment and Privatisation Project involving league champions Al Ittihad, Al Ahli, Al Nassr and Al Hilal.
PIF owns 75% of each of the four clubs, while their respective non-profit foundations own 25% of each of them.
