The 23-year-old, who scored the winning goal in Brazil’s recent 1-0 win over Honduras, is currently on international duty at the Copa America.
Firmino has scored three times in six appearances for Brazil and is likely to be a key figure for Dunga’s Brazil side in Chile.
He scored 10 goals and contributed 12 assists in the Bundesliga last season as Hoffenheim finished eighth and missed out on a place in the Europa League by just two points.
Several sources suggest Firmino is chasing a move from the club and that United have agreed a transfer for the player, whose versatility would fit into Louis van Gaal’s thinking.
He will undergo a medical at the Copa America this month and his transfer will be concluded once the tournament concludes, according to the latest reports.