The left-footer was released by Boston this summer after two seasons at York Street.
Despite starting his Pilgrims career well, Chapman struggled in his second season and was not offered a contract by returning manager Steve Evans.
Chapman, who came through the youth system at Blundell Park, has accepted the offer of a trial from 'Boro boss Nick Henry and is likely to feature in a number of pre-season games, possibly lining up against Town.