With back-up goalkeeper Shane Supple announcing his shock retirement from football last week, Roy Keane signed free agent Arran Lee-Barratt who took his place on the bench at West Brom, while Murphy will sign from League of Ireland side Bohemians later in the season.
Murphy, 26, started his career at Manchester City but failed to make a first-team appearance for the club and left in 2003. He then spent three years at Swansea City, making 13 appearances, before joining Bohemians in 2007.
Wright said: "Brian is coming in January so there will be competition for places which is what we need.
"I knew of Arran as a youngster when he was at the Academy but not too well.
"He has been training with us for the past three weeks or so and the surprise announcement gave Arran the chance to join the club.
"It fell into place for him and he was on the doorstep and he has been very good in training."