The Jacks are looking to bolster their attack after narrowly avoiding relegation last season, and could scoop the Belgian forward in the coming weeks.
The club were thought to be interested in the 20-year-old last summer, and are now looking to pursue a deal.
The news comes as key goalscorer Wilfried Bony is linked with a move away, with Arsenal, Liverpool and Everton reportedly keen on the Ivorian.
The same report suggests that Arsenal and Everton had looked at Batshuayi, while Portuguese heavyweights Porto are also looking to land a deal.
Regardless of the outcome of this particular deal, it appears new Swans manager Garry Monk is certainly trying to make signings that can lift the club back to the heights seen under Michael Laudrup.
Laudrup won Swansea their first ever major piece of silverware when the club won the League Cup in 2013, but the Dane was sacked less than a year later.
Monk took over the reins of the first-team, and was given the job on a full-time basis from Chairman Huw Jenkins. The new Chairman/manager partnership has seen Swansea linked with a host of names, including the likes of Tom Ince, Łukasz Fabiański and Jordy Clasie.
The Liberty Stadium attack is not the only area that could require recruitment this summer, with goalkeeper Michel Vorm another name being linked with a move away from the club. Vorm has been reported to be on Liverpool manager Brendan Rodgers' wish-list, as he looks to provide Simon Mignolet with some competition.
Batshuayi has been with Standard Liege first-team since the age of 17, scoring 37 times in 90 showings for the Belgian side.
His impressive form has already seen him called into the Belgium U21 squad, and could be just the type of player that could reignite the good times at Swansea.