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Aims. We study the close environment of nearby Cepheids using high spatialresolution observations in the mid-infrared with the VLTI/MIDI instrument, atwo-beam interferometric recombiner. Methods. We obtained spectra andvisibilities for the classical Cepheids X Sgr and T Mon. We fitted the MIDImeasurements, supplemented by B, V, J, H, K literature photometry, with thenumerical transfer code DUSTY to determine the dust shell parameters. We used atypical dust composition for circumstellar environments. Results. We detect anextended dusty environment in the spectra and visibilities for both stars,although T Mon might suffer from thermal background contamination. We attributethis to the presence of a circumstellar envelope (CSE) surrounding theCepheids. This is optically thin for X Sgr (tau(0.55microns) = 0.008), while itappears to be thicker for T Mon (tau(0.55micron) = 0.15). They are located atabout 15-20 stellar radii. Following our previous work, we derived a likelyperiod-excess relation in the VISIR PAH1 filter, f(8.6micron)[%]=0.81(+/-0.04)P[day]. We argue that the impact of CSEs on the mid-IRperiod-luminosity (P-L) relation cannot be negligible because they can bias theCepheid brightness by up to about 30 %. For the K-band P-L relation, the CSEcontribution seems to be lower (< 5 %), but the sample needs to be enlarged tofirmly conclude that the impact of the CSEs is negligible in this band.

Extended envelopes around Galactic Cepheids IV. T Monocerotis and X Sagittarii from mid-infrared interferometry with VLTI/MIDI