The Reds will pay about £22m (€30m) for the Hoffenheim player with add-ons likely to make the 23-year-old the club’s second most expensive signing in their history, after the £35m (€49m) purchase of Andy Carroll in January 2011.
Firmino, who will undergo a medical after his involvement in the Copa America is over, has agreed a five-year contract with the Reds after chief executive Ian Ayre flew out to Chile and finalised the deal.
The money for Firmino was outlined as planned summer transfer spending and the club do not have to sell Sterling, for whom Manchester City’s second £40m (€56m) offer was rejected earlier this month as they value him at over £50m (€70m), in order to balance the books.
They still have funds left for another centre-forward, with their sights set on Sevilla’s €28m-rated Colombia striker Carlos Bacca and Zenit St Petersburg’s Venezuelan Salomon Rondon, valued about £15m (€21m), who have both been in action at the Copa America.
Liverpool are remaining patient in their pursuit of Southampton’s England right-back Nathaniel Clyne, having already had a £10m (€14m) bid turned down, as the recruitment programme aimed at regaining Champions League football continues.
Meanwhile, Morgan Schneiderlin has indicated he could leave Southampton in the coming days.
The French defensive midfielder, 25, has been linked with a big money summer move away from St Mary’s Stadium, the likes of Manchester United, Arsenal and Tottenham all said to be possible destinations.
Schneiderlin joined the Saints from Strasbourg in June 2008.
The midfielder was back at his old club last night to help support a community event, and told France Bleu Alsace radio station he expected his future to be resolved shortly.
“The negotiations are ongoing and should be completed in the coming days, hopefully before the resumption of (Southampton’s pre-season) training next week,” he said.
Former Aston Villa striker Tony Cascarino has urged the club to cash in on Christian Benteke if he wants to leave.
Villa remain hopeful that the 24-year-old Belgium international, who reportedly has a £32.5m (€45m) buy-out clause in his current contract, will sign a new deal amid speculation that Manchester United and Liverpool could launch bids to land him this summer.
However, Cascarino has voiced concern over the player’s lack of form last season before the arrival of new manager Tim Sherwood, and believes he should be sold if his head has been turned.
The former Republic of Ireland international said: “You’re fortunate to play for clubs like Villa and I believe if the player doesn’t want to be there, then you have to look elsewhere.
“I don’t like keeping players who don’t want to play for you.
“He came back after injury and looked a shadow of the player for six months. He looked nowhere near the Benteke that ripped teams apart and took on teams on his own at times, and then he got back into the fold once Tim got the job and looked the Benteke of old. But that’s only over a four-month period.”
Elsewhere, defender Robert Huth has completed a permanent move to Leicester after a successful loan spell last season.