According to the Daily Star, Manchester United moved quickly after Firmino's brilliant performance for Brazil against Chile earlier this week, and have now reached an agreement with Hoffenheim for the 23-year-old.
The report goes on to state that Manchester United scout Peter Braund has been tracking Firmino's progress for some time and urged the club to make a move now as opposed to waiting for the inevitable bidding war which would come this summer.
German sports daily Bild felt that a deal was imminent for financial reasons, and Hoffenheim are now believed to be scouting Anderlecht forward Dennis Praet - according to Sportal via Belgium's Nieuwsblad - as Firmino's replacement for the 2015-16 season.
Firmino has scored six goals in 26 German Bundesliga appearances this season.
Manchester United have yet to make an official announcement on manager Louis Van Gaal's alleged most recent deal. If the signing is confirmed, of course, Firmino could remain at Old Trafford next season or be sent out on loan to further his development.