He is expected to be named in Spain's squad for the upcoming Euro 2024 qualifiers after scoring eight goals in 24 league games this season.
Sources have told ESPN that Newcastle United, AC Milan and Napoli are also closely following Veiga's progress. Leeds United and Bournemouth made an approach in January to lure the midfielder to the Premier League.
With a release clause set at €40 million, Premier League clubs see Veiga as a good signing not only for the present but also for the future.
Sources have told ESPN that Liverpool have commissioned reports on both Veiga's sporting and personal life.
Other sources have told ESPN that Veiga is on United's wish list for this summer, with the Manchester club wanting to strengthen their midfield options.
Borussia Dortmund's Jude Bellingham and Barcelona's Frenkie de Jong are still United's preferred choices but Veiga could be a good alternative considering the challenge that Erik ten Hag's side face to sign both their preferred options.
Sources have told ESPN that Ten Hag's top priority is to sign a striker but Veiga's arrival would be within the club's financial parameters.
Dortmund are set to command a fee in excess of €100m for Bellingham, who is under contract with the Bundesliga side until June 2025.
Celta, meanwhile, are trying to renew Veiga's contract, which runs until June 2026, to increase his release clause. However, it seems unlikely the player will accept as his last renewal did not meet the player's demands.
England seems to be the most logical destination for Veiga. However, the race for the midfielder has only just begun and other clubs, including Madrid, may join.
For the time being, Madrid have denied making any approaches to sign Veiga but have said they are monitoring the player just as they are keeping tabs other budding youngsters.