Aims. We aim to investigate the infrared excess of 45 Milky Way (MW) Cepheids combining different observables in order to constrain the presence of circumstellar envelopes (CSEs). Methods. We used the SpectroPhoto-Interferometry of Pulsating Stars (SPIPS) algorithm, a robust implementation of the parallax-of-pulsation method that combines photometry, angular diameter, stellar effective temperature, and radial velocity measurements in a global modelling of the pulsation of the Cepheid. We obtained new photometric measurements at mid-infrared (mid-IR) with the VISIR instrument at the Very Large Telescope complemented with data gathered from the literature. We then compared the mean magnitude of the Cepheids from 0.5 μm to 70 μm with stellar atmosphere models to infer the IR excess, which we attribute to the presence of a circumstellar envelope. Results. We report that at least 29% of the Cepheids of our sample have a detected IR excess (> 3 σ). We estimated a mean excess of 0.08 ± 0.04 mag at 2.2 μm and 0.13 ± 0.06 mag at 10 μm. Other Cepheids possibly also have IR excess, but they were rejected due to their low detection level compared to a single-star model. We do not see any correlation between the IR excess and the pulsation period as previously suspected for MW Cepheids, but a rather constant trend at a given wavelength. We also do not find any correlation between the CO absorption and the presence of a CSE, but rather with the stellar effective temperature, which confirms that the CO features previously reported are mostly photospheric. No bias caused by the presence of the circumstellar material is detected on the average distance estimates from a SPIPS analysis with a fitted colour excess. We also do not find correlation between the presence of IR excess and the evolution stage of the Cepheids. Conclusions. We report a fraction of 29% of Cepheids with an IR excess likely produced by the circumstellar envelope surrounding the stars. Longer period Cepheids do not exhibit greater excess than short periods as previously suspected from observations and theoretical dusty-wind models. Other mechanisms such as free-free emission, among others, may be at the origin of the formation of the CSEs. We also show that not fitting the colour excess leads to a bias on the distance estimates in our Galaxy.
