The Uruguayan striker's future at the club took a new twist on Wednesday when the striker dismissed quotes attributed to him in which he said he was willing to stay with the club.
The 26-year-old Uruguayan has been at odds with the Reds throughout the summer, going public with his desire to leave after claiming they reneged on a promise to allow him to go if they did not qualify for the Champions League.
Such has been his dissent amid two rejected bids from Arsenal that he has been told to train alone, with manager Rodgers saying he can only rejoin the first-team squad once he apologises for his actions.
However when asked for an update on the Suarez situation at his Thursday press conference, Rodgers refused to throw the reporters a bone.
"It's very simple. I want to concentrate on the players who are available. I won't be speaking any more on Luis," he said.
"There is nothing to be said that hasn't already been said. I am sure people are fed up of listening about it and reading about it.
"My only concentration is the players who are available and, from the first day they came back, they've had a real hard pre-season.
"Purely out of respect to them and the work which has been going on, I only want to talk about them."
Rodgers did talk briefly about potential signings says that "nothing was imminent" before adding that he still hopes to bring in one or two more signings before the end of the transfer window.
The Liverpool boss also said that he expects centre-halves Daniel Agger and Martin Skrtel to be fit for Saturday's Premier League opener against Stoke.