The 23-year-old is one of four players who have been told to find a new club by manager Stale Solbakken, formerly of Wolves, and Hearts are said to be nearing a deal.
Amankwaa was considered a rising star earlier in his career, having had trials with Ajax, Arsenal and Chelsea as a youngster.
Having decided to sign with Copenhagen he became a regular fixture in the side in the 2014/15 season, including playing on the continent as the club battled in the Europa League.
However, a series of injuries over recent seasons have curtailed his involvement in the first team. He will see his contract expire this coming summer.
The former Danish under-21 international is a right-sided winger with a lot of pace and strong dribbling abilities.
Tynecastle boss Craig Levein has previously stated his desire to sign a wide player during the January transfer window.
If Hearts manage to get the deal done, he’ll become the club’s third signing this month after the loan signings of Steven Naismith and Demetri Mitchell from Norwich City and Manchester United, respectively.