The 45-year-old recently agreed a contract extension until 2017 after he was linked with a move to Gillingham, who sacked Peter Taylor in December.
County have refused to comment but it is believed Edinburgh now wants to join the League One side.
Edinburgh succeeded Anthony Hudson as Exiles manager in October 2011.
The former Tottenham and Portsmouth defender led them back into the Football League after a 25-year absence.
Newport remain in the League Two play-off zone despite losing their last three games.
Heavy away losses to Cambridge United and Northampton Town were followed by Saturday's 1-0 home loss to league leaders Shrewsbury, a run in which Newport have failed to score a goal.
Gillingham have made an approach for Newport County manager Justin Edinburgh, BBC Wales Sport understands.