The 28-year-old played against Sunderland in a closed-door game at Riccarton on Saturday and will train with the squad throughout this week.
Comfortable at right-back or centre-back, Appiah previously played with the German club Arminia Bielefeld but spent the last two years in the Netherlands with NEC Nijmegen.
He was born in Schwelm in Germany to a Ghanaian father and an Italian mother.
Hearts want to recruit a versatile defender after Jordan McGhee joined Middlesbrough on loan. Head coach Robbie Neilson will decide whether Appiah is the man by the end of the week.
Northern Ireland internationalist Liam Donnelly, who spent last week on trial at Riccarton, has left after being told he would not be offered a contract.