The 27-year-old spent more than a decade at Rangers but never made a first-team appearance, despite completing 10 loan spells. His latest competitive action came with Airdrieonians in the 2024/25 Championship season.
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Wright has also featured for Livingston, Kilmarnock, Dumbarton, Alloa Athletic, Partick Thistle and Raith Rovers, with much of his experience coming in League One. He also spent pre-season on trial with Queen of the South and East Kilbride.
Hearts now have four senior goalkeepers, with Beau Reus currently first choice after replacing Alexander Schwolow. Ryan Fulton and Zander Clark provide additional cover, while Liam McFarlane and Harry Stone are on loan at Partick Thistle and Ayr United.