Carroll has been the subject of much speculation throughout the summer, with the Magpies and newly-promoted West Ham reportedly interested in acquiring the 23-year-old's services.
Newcastle are believed to be prepared to take Carroll on a season-long loan with a view to a permanent deal. However, it is a move that Rodgers is unlikely to sanction, with the Northern Irishman stating outright that he is not prepared to allow the Gateshead-born forward to return to his boyhood club.
"There is absolutely no chance [of a loan deal] – and certainly not Newcastle," Rodgers told reporters. "They got £35 million 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.
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"I have got [Luis] Suarez, Fabio Borini and Andy Carroll. I would need to be a nutcase to even consider at this moment to let Carroll go out, unless there are other solutions for that."
Carroll's first start of the Rodgers reign could come in this week's Europa League qualifier with Hearts, but the former Swansea boss revealed that the striker will have to undergo a scan before any decision on his fitness is made.
"He is due to start but he has felt a slight twinge, so we have had it looked at and scanned. We will see what his response is over the next 24 hours. If he is available and then fit, he will start," Rodgers added.