The study of open clusters stars is essential for our understanding of several open questions related to stars with planets and to planetary formation. We aim to investigate the dependence of planet formation on stellar mass and to compare in detail the chemical composition of stars with and without planets in a cluster of solar metallicity and age. Precise stellar radial velocities are used to search for massive (Jupiter masses or higher) exo-planets around stars of the Open Cluster M67. We observed a sample of 88 main-sequence stars, subgiants, and giants all with proper motion membership probability higher than 60%, using four telescopes and instrument combinations: the HARPS spectrograph at the ESO 3.6m, the SOPHIE spectrograph at OH?, the CORALIE spectrograph at the Euler Swiss telescope, and the HRS spectrograph at HET. 680 single observations, spanning a period of up to 8 years, have been recorded (Dec. 2011) and analyzed obtaining radial velocities with precision as good as similar or equal to 10 mis. After reducing all the observations to the HARPS zero point, the radial velocity (RV) measurements for each star are used to evaluate the RV variability along the cluster. We found 11 new previously unknown binary candidates, although the sample was pre-selected against binaries, other 11 stars clearly stand out with higher RV variability: 9 are interesting candidates for long term substellar companions, 2 stars present RV velocity variation too high to host a planet. The RV variance (including the stellar intrinsic variability and the observational error) for the bulk of stars is almost constant with stellar magnitude (therefore stellar gravity), at sigma =20 m/s. The high scatter of radial velocities, in excess to the measurement errors, may indicate that more stars are suitable candidates for hosting planets.