The integration of graphene into CMOS compatible Ge technology is in particular attractive for optoelectronic devices in the infrared spectral range. Since graphene transfer from metal substrates has detrimental effects on the electrical properties of the graphene film and moreover, leads to severe contamination issues, direct growth of graphene on Ge is highly desirable. In this work, we present recipes for a direct growth of graphene on Ge via thermal chemical vapor deposition (TCVD) and plasma-enhanced chemical vapor deposition (PECVD). We demonstrate that the growth temperature can be reduced by about 200 °C in PECVD with respect to TCVD, where usually growth occurs close to the melting point of Ge. For both, TCVD and PECVD, hexagonal and elongated morphology is observed on Ge(100) and Ge(110), respectively, indicating the dominant role of substrate orientation on the shape of graphene grains. Interestingly, Raman data indicate a compressive strain of ca. − 0.4% of the graphene film fabricated by TCVD, whereas a tensile strain of up to + 1.2% is determined for graphene synthesized via PECVD, regardless the substrate orientation. Supported by Kelvin probe force measurements, we suggest a mechanism that is responsible for graphene formation on Ge and the resulting strain in TCVD and PECVD.