This is according to leading Brazilian website Globo Esporte, whose reporters Marcelo Hazan and Richard Souza - currently in Chile for the Copa America - understand Liverpool can wrap up the deal by offering €25 million.
Firmino will leave his current employers for an English club this summer, according to recent reports in Germany, but the 23-year-old could now be headed to Anfield instead of Old Trafford.
Chelsea and Arsenal have also been linked with the versatile South American in recent weeks, but Liverpool appear to be pushing hardest to bring the player to the Premier League.
Brendan Rodgers is acutely aware of the need to strengthen his squad in time for the 2015-16 campaign after last season’s struggles.
Jurgen Klopp is lurking
Liverpool finished the 2014-15 season in sixth position, leaving Rodgers fighting for his job earlier this summer.
Rumours were rife that Liverpool chiefs would attempt to replace their current manager with Jurgen Klopp, who left Borussia Dortmund at the end of last season, but that plan had to be scrapped after the coveted German coach announced his intention to take a six-month break.
The Reds have subsequently stuck by Rodgers, but the situation could change again if performances and results fail to improve during the opening months of the upcoming season.
That Klopp has not yet struck an agreement with another club and would relish the opportunity to manage a Premier League team must make him an extremely attractive proposition for the Liverpool hierarchy.
Rodgers has to get his next signings right
But Rodgers will at least have the opportunity to prove, once again, that he’s the best man to take Liverpool forward.
The Northern Irish coach failed to make the most of the £75 million generated by the sale of Luis Suarez - the £16 million acquisition of Mario Balotelli, in particular, was a disaster - but Firmino could represent good value for money.
The former Figueirense star, who has scored four goals in nine appearances for the Brazilian national team, has demonstrated that he’s capable of cutting it in a major European league with Hoffenheim over recent seasons.
With 49 goals in 153 matches, Firmino is not a prolific forward, but is someone with the ability to bring the best out of his teammates.
Liverpool fans: would you be happy if Brendan Rodger signed Roberto Firmino this summer? Have your say by leaving a comment in the box below!