Carroll has struggled for form and fitness since his record-breaking £35m move to Liverpool and he has been repeatedly linked with a return to the North East.
Newspaper reports on Wednesday suggest that Alan Pardew's side have rejected two proposals from Liverpool but are open to a £10m deal for the 23-year-old.
A Newcastle source told The Sun: "Andy's a Geordie boy and it would be great to bring him home.
"But while he left on his terms he'd have to come back on ours."
Carroll has scored six goals in 31 games for Liverpool.