In this study, the rheological and melt-flow behavior of polyether ether ketone (PEEK) filled with multi-walled carbon nanotubes (MWCNTs) was investigated to evaluate its melt-spinnability. A rheological percolation threshold was detected below 1 wt% MWCNT, accompanied by a significant increase in storage modulus (G′) and loss modulus (G″), as well as a decrease in complex viscosity (η*), indicating the formation of a nanotube network and a transition from viscous to elastic behavior. Neat PEEK (n = 0.09, m = 1.16) and PEEK containing 1 wt% MWCNT (n = −0.33, m = 0.66) exhibited stable flow behavior and remained within the spinnable range due to moderate shear-thinning characteristics. At 2 wt% MWCNT, increased elasticity and partial network formation (n = −0.89; m = 0.39) resulted in reduced spinnability. Above 2 wt%, the materials were not spinnable due to their highly elastic behavior (m < 0.3) and thread breakage during spinning. Van Gurp–Palmen and Cole–Cole analyses confirmed that PEEK/MWCNT blends with phase angles δ > 40° and MWCNT contents below 2 wt% are spinnable. The optimum MWCNT content for stable melt spinnability was determined to be 1 wt%, though 2 wt% was also spinnable under certain spinning conditions.