Miguel Herrera has been fired as Mexico coach following allegations that he punched a television reporter.
Former Arsenal midfielder Abou Diaby has rejected an offer to join West Brom in favour of finding a new club in France.
The former Cardiff City forward joined Championship side Blackpool in March and is now available on a free transfer after his short-term deal at Bloomfield Road expired.
Despite Earnshaw’s availability, Edinburgh believes Newport would struggle to persuade the 33-year-old to join the League Two club for next season.
“There’s no truth in that story,” Edinburgh told the South Wales Argus.
“I’m not sure we’d be able to attract someone of his stature, but there’s nothing wrong with us being linked to players like that.
“It shows how far the club has come in recent years, but I’ve got plenty of strikers here already.”