West Ham and Arsenal are also tracking the Colombian, but Barcelona have now entered the race.
Barcelona appear to be struggling to sign a striker, with many forwards being put off the Camp Nou because of the difficulty in being promised regular first-team football.
Any forward who joins the Camp Nou would have to find a way into the team ahead of Lionel Messi, Luis Suarez and Neymar, something that appears near impossible.
For this reason, Bacca may be a good fit at Barca. Approaching 30 years old, the AC Milan striker may be happy to sit out a few games. On top of this, he has also stalled on West Ham's offer because of his desire in the Champions League, something that Barcelona can offer.