The 21-year-old has quickly become one of the most sought-after talents in La Liga after a breakthrough season with Celta during the 2022-23 campaign.
Veiga scored 11 goals and set up four more in 36 La Liga outings and was subsequently called up to the Spain Under-21 squad for the recent European Championships, where they finished as runners-up to England.
The exploits of Veiga - who can operate as a central midfielder and number 10 - have understandably led to interest from several of Europe's top teams, including the likes of Paris Saint-Germain and Chelsea.
However, speculation surrounding the Spaniard's future has cooled in recent days, as Chelsea are instead pursuing a move for Rennes' Lesley Ugochukwu, while PSG are working to finalise the signings of Ousmane Dembele and Goncalo Ramos.
Hoping to capitalise on the situation, Cope - via The Mirror - claims that Liverpool have now come forward with an offer of €25m (£21.5m), which falls well below Celta's asking price.
The La Liga side's head coach Rafael Benitez - a former Liverpool manager - is supposedly reluctant to let Veiga leave for anything less than €40m (£34.5m), as his contract runs until 2026.
However, the report adds that Liverpool are offering Celta the chance to keep Veiga on loan for the 2023-24 season, which they hope will make up for the cut-price fee.
Veiga has risen through the Celta youth system to make 55 appearances for the club in all tournaments, and all of his goal contributions for Benitez's side so far came in the 2022-23 La Liga campaign.
Liverpool have undergone a significant period of change in the centre of the park this summer, waving goodbye to Fabinho, Jordan Henderson, Naby Keita, James Milner, Leighton Clarkson, Alex Oxlade-Chamberlain and Juventus loanee Arthur.
Thiago Alcantara is also in the final year of his contract, and the Spaniard is expected to walk out of the door on a free transfer next summer if he is not sold in the coming weeks.
The Reds' midfield reshuffle has seen Dominik Szoboszlai and Alexis Mac Allister arrive at Anfield for a combined £95m, and Klopp is now expected to spearhead a search for Fabinho's successor in the number six role.
Southampton's Romeo Lavia is still believed to be high on the Reds' shortlist, but the Saints have already rejected a £37m bid from Liverpool and have slapped a £50m price tag on the 19-year-old.
Bayern Munich's Ryan Gravenberch and Manchester City's Kalvin Phillips are reportedly seen as alternative options, but Fiorentina's Sofyan Amrabat - another rumoured target - is apparently edging closer to a £30m move to Manchester United.
Liverpool suffered a 4-3 defeat to Bayern Munich in their penultimate summer friendly on Wednesday and cap off their preparations for the new term against Darmstadt 98 at Deepdale on Monday.