The Daily Record says Hearts look set to close a deal to land Northern Ireland star Washington.
Jambos first-team coach Austin McPhee has been trying to convince the Sheffield United star to move to Tynecastle since the end of the season.
There has been significant progress over the last 48 hours and Hearts are confident they can get now get a deal over the line.
The 27-year-old is set to become a free agent after being told his Bramall Lane won't be renewed, following their promotion to the English Premier League.
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