A strong reduction of the Curie temperature TC has been observed for room-temperature-grown fcc Fe films on Cu(001) when covered with 1 monolayer (ML) Cu for all Fe thicknesses up to the fcc-bcc transition of the Fe film at approximate to 11 ML. At 2 ML Cu coverage this decrease of TC partially recovers and approaches a constant lower value on further increasing Cu coverage. The correlation of this observed magnetic behavior with electronic and possible structural changes of the Fe film upon Cu coverage is discussed.