Carroll has been linked with a loan move back to Newcastle. The England striker joined Liverpool in a £35million deal from Newcastle in January 2011.
"There is absolutely no chance [of a loan deal] - and certainly not Newcastle," Rodgers said.
"Newcastle got £35m for this player last year. To even consider wanting to take him on loan is a liberty really.
"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."