BILD says despite signing the Dutchman on deadline day, Newcastle managed to negotiate an outright option in the loan arrangement.
Newcastle have paid €1m for his season-long loan with a €11.5m option agreed to buy him permanently.
The 22 capped international has played five times for Newcastle in the Premier League this season and scored in defeat to Liverpool.
In the last game before international week, at home against Manchester United (1-0 win), he was on the field for ninety minutes.
Willems has a contract with Eintracht until mid-2021.