Wright is the second goalkeeper to arrive at Rodney Parade this summer, with ex-Northampton Town stopper Nik Tzanev due to complete his move to Wales following international duty with New Zealand.
A former Nottingham Forest trainee, 6ft 5in Wright made his EFL breakthrough at Lincoln, playing 32 times for the Imps before moving to Grimsby.
But he was made available for transfer with a year remaining on his contract, opening the door for a move to Newport.
The Exiles had been keen to add to their goalkeeping options after long-serving stopper Nick Townsend departed after seven years, turning down a new contract to sign for National League Eastleigh.
Newport have made significant progress in their summer recruitment, including the signing of Wales midfielder Matt Smith after his departure from Scottish outfit St Johnstone.
As well as Tzanev, Newport have brought in Barrow striker Ged Garner, Haverfordwest County centre-back Lee Jenkins and Michael Reindorf on loan from Cardiff City, while defender James Clarke has signed a new deal.
