We report the discovery of gas streaming motions along nuclear spiral arms toward the LINER nucleus of the galaxy NGC 6951. The observations, obtained using the GMOS integral field spectrograph on the Gemini North telescope, yielded maps of the flux distributions and gas kinematics in the Hα, [N II] λ6584, and [S II] λλ6717, 6731 emission lines of the inner 7''×15'' of the galaxy. This region includes a circumnuclear star-forming ring with a radius of ~500 pc, a nuclear spiral inside the ring, and the LINER nucleus. The kinematics of the ionized gas is dominated by rotation, but subtraction of a kinematic model of a rotating exponential disk reveals deviations from circular rotation within the nuclear ring that can be attributed to (1) streaming motions along the nuclear spiral arms and (2) a bipolar outflow that seems to be associated with a nuclear jet. On the basis of the observed streaming velocities and geometry of the spiral arms, we estimate a mass inflow rate of ionized gas of ~3×10-4 Msolar yr-1, which is on the order of the accretion rate necessary to power the LINER nucleus of NGC 6951. Similar streaming motions toward the nucleus of another galaxy with a LINER nucleus, NGC 1097, have been reported by our group in a previous paper. Taken together, these results support a scenario in which nuclear spirals are channels through which matter is transferred from galactic scales to the nuclear region to feed the supermassive black hole.