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The champions, who are also out of the League Cup, are 15th in the table following Saturday's 3-1 home loss.
Asked if he thought it was his last game in charge, he said: "No, I don't."
The 52-year-old told BBC Sport: "The fans aren't stupid. They know how much myself and the players are trying."
Listen: I’ll turn Chelsea around if they allow it – Mourinho
Listen to all the goals from Saturday's game
Mourinho has never lost more than six league matches in an entire league campaign as Chelsea manager, and says both he and the team will continue to battle.
"The fight goes on, but sometimes there are fights very impossible to win. You go to a fight with different ammunition," he added, having been photographed in discussion with his coaches on the pitch after the match.
Chelsea made the perfect start when Ramires headed them into the lead. But two goals from Philippe Coutinho and a third from Christian Benteke gave Jurgen Klopp's side the three points.
In the BT Sport interview, the Portuguese manager looked visibly shaken and responded with questions about his team's performance and his future by repeating, "I have nothing to say".
And in his BBC interview, when asked about Liverpool midfielder Lucas not receiving a second yellow, Mourinho responded with, "I don't speak" before adding in his news conference later that he feared he would be reprimanded if he did express his views.
"I am punished if I tell you," he said, as he awaits the outcome of two disciplinary matters.
He then cryptically added: "And what happened in the second half, everything was the consequence of a crucial moment. Everybody in the stadium saw, the players felt it. What happens is now a consequence."
Mourinho said his relationship with the press would change in the build up to last week's defeat against West Ham. It was in reaction to the reporting of him being filmed by members of the public using mobile phones.
In Saturday's news conference, he added: "As professionals the players are not getting the respect they deserve. We are not going to have a great Saturday dinner.
"I am sorry about lack of respect for professionals. You are intelligent guys. At the next press conference I will bring you nice glasses and you will see things in a better way."
BT Sport reporter Des Kelly: Jose, a 3-1 defeat after a fabulous start to the game, the game just got away from you then?
Mourinho: I have nothing, nothing to say.
Kelly: Nothing to say about the game at all?
Mourinho: Nothing, nothing to say.
Kelly: Nothing to say about the Lucas (Leiva) decision that left him on the pitch?
Mourinho: Nothing to say, I have nothing to say.
Kelly: The Diego Costa clash?
Mourinho: Nothing to say, I am so sorry, I have nothing to say.
Kelly: Do you not think it's time to have a chat to the fans to give them some message, an indication of your thinking?
Mourinho: They are not stupid.
Kelly: We heard them chanting your name.
Mourinho: The fans are not stupid.
Kelly: You did say before this game that you were not worried, are you a little bit more worried now?
Mourinho: Worried about what?
Kelly: Your future, your own position at the club, the backing of the board?
Mourinho: No.
Kelly: Nothing about the game at all?
Mourinho: No.
Kelly: No individual performance you would like to pick out?
Mourinho: No.
Kelly: Nothing about the performance going ahead?
Mourinho: Nothing.
Kelly: Nothing about the way we could fix it?
Mourinho: I cannot say.
Kelly: Thank you for your time Jose.
Manager Jose Mourinho believes he will be given time to turn around Chelsea's fortunes after Liverpool fought back from a goal down to inflict a sixth Premier League defeat this season.