The 21-year-old's future is currently clouded in uncertainty following his impressive exploits in the 2022-23 season, where he registered 11 goals and four assists in 36 La Liga matches.
Veiga's contract with Celta Vigo still has another three years to run, but the Spain Under-21 international's terms reportedly include a release clause set at €40m (£34.3m).
Veiga is believed to have attracted glances from some of the Premier League's Big Six clubs, with all of Chelsea, Man City and Liverpool weighing up a move.
Chelsea apparently view Veiga as an ideal successor to Mason Mount, who is on the verge of signing for Manchester United, while Mateo Kovacic, Ruben Loftus-Cheek and N'Golo Kante's exits have also left the Blues thin on the ground in midfield.
Meanwhile, Man City captured Kovacic to replace Barcelona-bound Ilkay Gundogan, but the futures of Kalvin Phillips and Bernardo Silva are also being called into question.
Liverpool are also thought to be after one more midfield signing following the arrivals of Dominik Szoboszlai and Alexis Mac Allister, while Barcelona have also been credited with an interest in Veiga.
However, The Guardian claims that PSG have also joined the race to sign the much-coveted midfielder and have quickly accelerated their efforts to bring him to the Parc des Princes.
The Ligue 1 champions are yet to trigger Veiga's release clause, though, and the report adds that Chelsea are also expected to make their move in the coming days, but Romeo Lavia and Moises Caicedo are currently seen as more of a priority.
PSG underwent a significant midfield revamp last summer with the signings of Renato Sanches, Carlos Soler, Vitinha and Fabian Ruiz, but the quartet all struggled to impress during their debut campaigns.
Amid talk of a possible move away for Saudi Arabia-linked Marco Verratti, PSG - who are set to confirm Christophe Galtier's dismissal and Luis Enrique's arrival as head coach this week - are after further additions in the engine room.
The Ligue 1 champions are expected to confirm the arrival of Manuel Ugarte from Sporting Lisbon at some point in July, while Milan Skriniar and Marco Asensio will also be unveiled shortly.
Benfica starlet Cher Ndour and Mallorca playmaker Lee Kang-in are also poised to join PSG's ranks for the 2023-24 season, while Enrique is believed to have earmarked Man City's Silva as another priority target.
Veiga came through the youth system at Celta Vigo and has now turned out for the senior team on 55 occasions in all tournaments, making his debut as an 18-year-old in September 2020.
The midfielder - who is competent as a number six or number eight - is currently on duty at the Under-21 European Championships with Spain, who face Ukraine in Wednesday's semi-final for the chance to meet England or Israel in the final.