Rodgers hit out at Newcastle for trying to get Carroll - a player they sold to Liverpool for £35 million in January 2011 - back on the cheap, and said he did not have enough depth in his squad to let the England striker leave without raising funds for a replacement.
"There is absolutely no chance [of a loan deal] - and certainly not Newcastle," Rodgers said, quoted by the Guardian. "Newcastle got £35m for this player last year. To even consider wanting to take him on loan is a liberty really."
He went on: "We have got a very small squad as it is. We have lost a lot of players this summer and I have not replaced them, as of yet. That is the reality of where we are.
"I need a minimum of three strikers. Once the window shuts, that is it until January. I have got (Luis) Suarez, Fabio Borini, and Andy Carroll.
"I would need to be a nutcase even to consider at this moment to let Carroll go out, unless there are other solutions for that."
Newcastle and other clubs were alerted to Carroll's potential availability in July when Rodgers suggested he would be open to a loan move for the 23-year-old if it "can benefit the club in the long term".
However, the offer made by Newcastle - reported to be a loan move with a view to a £12million permanent return to Tyneside with Liverpool still responsible for a sizeable chunk of his wages in the meantime - has particularly irked Rodgers.
Meanwhile, Rodgers indicated he did not expect any further developments in Manchester City's pursuit of defender Daniel Agger.
"I can't see anything at this stage to be honest," Rodgers said. "He is not a player that I want to lose. Like every player in the group, if someone comes in with a big, big offer then that might be out of my hands in relation to the club and where it is at with finances. I have never thought that I was going to lose Daniel. Secondly, he doesn't want to go."