The 23-year-old Brazil international’s ‘long-term’ deal is subject to a medical, which will take place when he finishes his commitments at Copa America, underway in Chile.
The Reds are likely to initially pay £25m to the Bundesliga club and add-ons could take the sum close to £29m, which will make the Brazilian club’s second most expensive signing.
The Merseyside club had signed striker Andy Carroll from Newcastle United for a club record fee of around £35m in 2011.
He scored 47 goals in 151 appearances during his four-and-a-half season stay at Hoffenheim and had two years left on his contract with the German club.
Last term, he bagged seven goals in 33 Bundesliga appearances.
Firmino has won nine Brazil caps to date since his full debut in November 2014 – netting four goals, including one in the Seleçao’s 2-1 victory over Venezuela on Sunday, which advanced them into the Copa America quarterfinals.
The striker is Anfield club’s fifth signing in this summer transfer window after James Milner, Adam Bogdan, Danny Ings and Joe Gomez.
Liverpool have so far turned down two bids (of £30m and £40m) from City for Raheem Sterling but Firmino’s arrival could see the Citizens prepare a third offer, of around £50m, for the England winger.