We present a method for calculating the asymptotic shape of interacting vortex filaments in incompressible Euler flows using delay differential equations. Neglecting the filaments’ coresize, the asymptotic shape of the filaments is self-similar up to logarithmic corrections, albeit non-universal. We demonstrate explicitly that the asymptotic geometry of the collapse of two interacting fil- aments depends on the pre-factor of the scaling law of their separation distance, the angle between the tangent vectors at their approaching tips, and the ratio of their circulations. We then explore the validity of the filament approximation in the limit of approaching the singularity. We show that a sufficiently fast stretching-rate to maintain this approximation is inconsistent with all collapse geometries. This suggests that a singular solution to the Euler equations based on stretching of vortex filaments is unlikely to exist for any initial conditions.