Context. The knowledge of the present-day mass function of young clusters and the mass of their coolest substellar members is essential to clarify the brown dwarf formation mechanism, which still remains a matter of debate. Aims. We searched for isolated planetary-mass T-dwarfs in the ∼3 Myr old Serpens Core cluster. Methods. We performed a deep imaging survey of the central part of this cluster using the WIRCam camera at the CFHT. Observations were performed through the narrow-band CH 4 off and CH 4 on filters, to identify young T-dwarfs from their 1.6 μm methane absorption bands, and the broad-band JHK S filters, to better characterize the selected candidates. We complemented our WIRCam photometry with optical imaging data from MegaCam at CFHT and Suprime-Cam at the Subaru telescope and mid-infrared flux measurements from the Spitzer "core to disk" (c2d) Legacy Survey. Results. We report four faint T-dwarf candidates in the direction of the Serpens Core with CH 4 on−CH 4 off above 0.2 mag, estimated visual extinction in the range 1−9 mag and spectral type in the range T1−T5 based on their dereddened CH 4 on−CH 4 off colors. Comparisons with T-dwarf spectral models and optical to mid-infrared color-color and color-magnitude diagrams, indicate that two of our candidates (ID 1 and 2) are background contaminants (most likely heavily reddened low-redshift quasars). The properties of the other two candidates (ID 3 and 4) are consistent with them being young members of the Serpens Core cluster, although our analysis can not be considered conclusive. In particular, ID 3 may also be a foreground T-dwarf. It is detected by the Spitzer c2d survey but only flux upper limits are available above 5.8 μm and, hence, we can not assess the presence of a possible disk around this object. However, it presents some similarities with other young T-dwarf candidates (S Ori 70 in the σ Orionis cluster and CFHT_J0344+3206 in the direction of IC 348). If ID 3 and 4 belong to Serpens, they would have a mass of a few Jupiter masses and would be amongst the youngest, lowest mass objects detected in a star-forming region so far.