Sevilla left-winger Ruben Vargas has so far scored two goals and provided an assist to help his country qualify for the round of 16 tie against Colombia.
He turned down a move to Trabzonspor, although it would have offered him a pay rise ahead of the World Cup, and he is understood to be keeping any thoughts about his future private while in the US.
But the ABC Seville newspaper and website, which closely follows his club, are reporting interest from the Seagulls in the striker, who turns 28 on August 5.
They believe he is happy at Sevilla and if he leaves, it will be to a team where he can play in Europe.
He had three goals and six assists last season as his club struggled in the bottom half of La Liga.
The ABC understands Albion is “closely monitoring its availability”.
Besiktas are also believed to be interested in Vargas, as well as ex-Albion striker Leandro Trossard at Arsenal.
Sevilla made it clear last season that they would not let the player go for less than £11 million, a figure that is likely to rise during the World Cup.
But like most clubs across Europe, they will be receptive to the riches of the Premier League.
Switzerland head coach Murat Yakin said: “He can create numerical superiority with his dribbling because he is fast and dynamic in his play.”
“He is also very dangerous in front of goal and has a great shot.
“In addition, Ruben also brings a lot of defensive elements that help the whole team, and that is why he is such an important player for our team.”
Vargas was born in Switzerland to a Dominican father and made a name for himself in the Bundesliga with Augsburg.
He has three years remaining on his current contract with Sevilla and is one of several players Albion have been linked with over the busy summer ahead.