The Reds star is currently tied down until 2020.
And since joining from Hoffenheim two years ago, the £29m signing has progressed at Anfield.
Despite arriving as a versatile forward, Firmino has now established himself as Liverpool’s first-choice striker.
To rewards his efforts, the Liverpool Echo claim he is next in line for a new deal.
The club want to secure the Brazil international’s long-term future.
Klopp in particular is said to view Firmino as key to his plans looking ahead, with Liverpool’s hierarchy in agreement.
The Reds want to fend off interest from elsewhere and ensure he does not fall down a similar path to Emre Can, who could leave for Juventus in 2018 with his deal running down.
Meanwhile, the Liverpool Echo add a new contract for Ben Woodburn is also high on their agenda.
The Wales international is set for improved terms when he turns 18 later this month.
Woodburn has made 10 senior appearances for Liverpool under Klopp.
With the young forward seen as one for the future on Merseyside, Firmino remains important for the here and now.
The 26-year-old has two Premier League goals in seven appearances and one in the Champions League so far this term.
Despite being benched for the 1-1 draw at Newcastle before the international break, Firmino is expected to start at home to arch-rivals Manchester United next Saturday (12.30pm).
