Confinement in fiber traps with two optical fibers facing one another relies on balancing the optical forces originating from the interaction of a scattering micro-object with the light beams delivered through the fibers. Here we demonstrate a novel type of dual fiber trap that involves the use of nanobore fibers, having a nano-channel located in the center of their fiber cores. This nano-element leads to a profound redistribution of the optical intensity and to considerably higher field gradients, yielding a trapping potential with greatly improved tuning properties compared to standard step-index fiber types. We evaluate the trap performance as a function of the fiber separation and find substantially higher stiffness for the nanobore fiber trap, especially in the range of short inter-fiber separations, while intermediate distances exhibit axial stiffness below that of the standard fiber. The results are in agreement with theoretical predictions and reveal that the exploitation of nanobore fibers allows for combinations of transverse and axial stiffness that cannot be accessed with common step-index fibers.