The Baggies have enquired about signing the 23-year-old, who scored 18 goals last season and is valued at £18 million.
Borja joined Atletico's academy at the age of four but has made just a single La Liga appearance for the first team, that occasion coming six years ago,
However, the 23-year-old has enjoyed a series of highly successful loan spells across Spain since stepping up to senior level and has built a reputation as one of the country's greatest prospects.
The striker scored 23 goals in 38 Segunda Division games with Real Zaragoza in 2014-15, and then hit 19 in all competitions for La Liga side Eibar last campaign, form which has seen him tipped as the next big thing in his home country.
Several clubs are thought to be interested in being the next club to continue his development, and according to the Daily Mail one of those is West Brom, with Tony Pulis thought to be a fan of the talented star.
Pulis is keen to boost his attack as the club expect Saido Berahino to leave this summer, and veteran forward Rickie Lambert was linked with a move to Aston Villa earlier this week.
Borja would be a gamble, having only played a single season in a top division to date and never having played outside his native Spain.
What's more, Atletico are thought to rate their hottest prospect at around £18 million, a huge sum for a player who has spent the vast majority of his career in the Spanish second tier.
However, he clearly has the talent and the strike rate to be a success if he could adapt to the English game, and the Baggies desperately need more numbers and quality in an attack which is likely to consist of just Salomon Rondon and Jonathan Leko.