The Reds know a win or at least matching Arsenal's result against Everton on Sunday will be enough to secure a top-four Premier League finish and claim Champions League football for next season.
Firmino missed last week's 4-0 win over West Ham with a minor muscle injury but Klopp is hopeful the Brazil international will prove his fitness in the next 24 hours and be available to take on the already-relegated Boro.
When asked about the chances of Firmino being involved, Klopp told reporters in his press conference: "(There's) no decision until now but he's good. He's not been part of team training yet but we will make a final decision tomorrow."
If Firmino doesn't make it, the Reds are likely to be unchanged with Daniel Sturridge again expected to start up front and Philippe Coutinho likely to be deployed in a deeper role like against the Hammers.
Jordan Henderson and Sadio Mane remain on the sidelines.
Klopp knows it's vital his side do not fall at the final hurdle in their bid to make the top four and wants them to start the game strongly.
He added: "It's important. So far it's been a very successful season. We have to perform and win. The best thing would be if we are 4-0 up after two minutes but that won't be possible. Starting well is very important."