Fused filament fabrication technique is the most popular additive manufacturing that has received extensive attention, revolutionizing industrial production processes. In addition to the standard thermoplastic polymers, much emphasis is placed on developing electrically conductive functional filaments being the most interesting. While there are conductive filaments that are commercially available, the market lacks a wide range of flexible options. Hence, this paper presents innovative filaments based on the combination of thermoplastic hot melt copolyester adhesive (HMA) with multi-walled carbon nanotubes (MWCNT). The incorporation of carbon nanotubes into the HMA was carried out through a two-step process. First, a masterbatch of 10 wt. % MWCNT was diluted with pure polymer using a half-industrial twin-screw extruder to obtain concentrations in the range of 1–9 wt. %. Consequently, the nanocomposite pellets were extruded again into the form of filaments. The rheological analysis demonstrates that adding MWCNT to the HMA increases both the viscous and elastic behavior of the composites. The homogenously dispersed nanotubes in the polymer matrix led to electrical conductivity of 1.39 S/m for the filaments containing 10 wt. % MWCNT. They are also characterized by the stiffness and tensile strength of about 300 and 13 MPa, respectively. With high thermal stability up to 360 °C, low porosity, and high flexibility, the developed filaments are suitable for 3D printing. The printability of all filaments was confirmed, exhibiting lack of breakage during printing and visibly better quality of the parts with the higher nanotube content.