The Reds boss is unhappy with the attitude shown by the Uruguay international - who seems intent on forcing a move to Arsenal - in recent weeks and has been told to train on his own.
Rodgers rejected Suarez's claim he had reneged on an agreement last summer which would allow the player to leave for a Champions League club, saying interviews given by the 26-year-old suggesting as much showed "total disrespect" to the club.
However, Suarez remains determined to find a way out of Anfield and admits even captain Steven Gerrard's pleas for him to stay cannot offset the lure of the Champions League, while also reiterating he would jump at the chance to go to Real Madrid.
"I'm still defending the Liverpool shirt and I will do so until the last minute," Suarez, who was left at home for Wednesday night's friendly against Valerenga in Norway because of injury and is unlikely to feature against Celtic in Dublin on Saturday, told Spanish newspaper Marca.
"My aim is to reach an agreement with the club in a friendly manner and I hope the clause we agreed a year ago is fulfilled. Up to now, it hasn't been.
"It's not that I want to or am desperate to leave, but I like things to be clear and a year ago I made my position to the club clear.
"Gerrard has a lot of history with Liverpool and his words are very important but everyone should understand that I need to play the Champions League to keep growing as a footballer.
"Now I have the chance, I want to take it.
"A Champions League team (Arsenal) is making a big effort to get me and one always wants to be where they feel comfortable.
"There were a lot of rumours (about Real) that were different to reality: Madrid never called me personally or Liverpool to try and negotiate.
"Every player aspires to reach the top and Real Madrid is among the most important clubs in the world. It would be very difficult to say no to Real Madrid."
Pictures of Suarez leaving Liverpool's training ground, apparently in tears, subsequently emerged on Thursday.