Talks between Damour and Hearts progressed significantly throughout the course of Thursday, and he is poised to move to Edinburgh with a pay-off for the final year of his Cardiff contract.
The two clubs had agreed a season-long loan for the Frenchman but he would prefer a permanent transfer. Barring an unexpected late hitch, talks should be completed before Saturday's Premiership match with Ross County at Tynecastle.
That would make Damour manager Craig Levein's fifth recruit of the summer after Steven Naismith, Jamie Walker, Conor Washington and Craig Halkett.
The 28-year-old is a former French youth internationalist who represented his country at every age level between under-16 and under-20. He graduated from the famous Clairefontaine football academy in France aged 17 to join Strasbourg.
He spent time with a number of clubs in France and Belgium before moving to Cardiff in 2017. Damour was a key figure in City's promotion to the English Premier League in 2018 but has since fallen out of favour under manager Neil Warnock.
Hearts moved to sign him after learning midfielder Peter Haring would be out until next month with a pelvic injury.