The 21-year-old endured a breakout campaign in 2022-23, contributing 11 goals and four assists from his 36 appearances in La Liga.
As a result, Veiga has also gone on to earn nine caps for Spain Under-21s, while a release clause in his contract has led to widespread interest this summer.
Moving to a Premier League club appeared a foregone conclusion at one stage with the likes of Manchester City, Liverpool and Newcastle United all being heavily linked.
However, it emerged earlier this month that Napoli had held positive discussions with Veiga, who appeared enthusiastic about joining the Serie A champions.
A switch to Naples would have represented a significant coup for the Italian giants, yet it now appears that he will be instead plying his trade in the Saudi Pro League.
According to transfer expert Fabrizio Romano, Veiga has agreed to sign for Al-Ahli and is expected to finalise a transfer to the Middle East in due course.
Romano describes Veiga as a "secret top target" for the Saudi giants, a former top-flight winner who earned promotion from the second tier in 2022-23 and are owned by the Public Investment Fund (PIF).
Despite their status last season, Al-Ahli are one of the favourites to win the league title this term courtesy of the amount of signings that have been made by club officials and Matthias Jaissle, who was allegedly key in negotiations with Veiga.
Goalkeeper Edouard Mendy, defenders Roger Ibanez and Merih Demiral, midfielder Franck Kessie, and attackers Allan Saint-Maximin, Riyad Mahrez and Roberto Firmino have all signed for The Royal.
While Saudi clubs are able to have eight foreign players in their ranks, the other PIF-controlled clubs - Al-Ittihad, Al-Nassr and Al-Hilal - are only able to register five for continental competition.
That ruling may influence how the aforementioned three clubs do business in the remaining days of the transfer window, although they are under pressure to match Al-Ahli from a domestic standpoint.
Al-Ahli are one of three clubs to have recorded maximum points from their opening two fixtures of the new Saudi Pro League campaign.