The 24-year-old wants to leave Barcelona this January after falling down the pecking order.
The striker arrived at the Camp Nou from Valencia in the summer of 2016 for over £20million but he has made just two league appearances this season.
Southampton are keen on signing Alcacer in January and they are ready to give him the first-team football he craves.
But, according to Onda Cero, Celta Vigo are trying to sneak in and beat the Saints to the Spaniard's signature.
The LaLiga side are in the market for a forward too and they believe they could win the race for Alcacer by offering him the chance to stay in his home country.