We report on very long baseline interferometry (VLBI) observations of the radio emission from the inner region of the Crab nebula, made at 1.6 GHz and 5 GHz after a recent high-energy flare in this object. The 5 GHz data have provided only upper limits of 0.4 milli-Jansky (mJy) on the flux density of the pulsar and 0.4 mJy/beam on the brightness of the putative flaring region. The 1.6 GHz data have enabled imaging the inner regions of the nebula on scales of up to ≈ 40′′. The emission from the inner "wisps" is detected for the first time with VLBI observations. A likely radio counterpart (designated "C1") of the putative flaring region observed with Chandra and HST is detected in the radio image, with an estimated flux density of 0.5 ± 0.3 mJy and a size of farcs 2-0 farcs 6. Another compact feature ("C2") is also detected in the VLBI image closer to the pulsar, with an estimated flux density of 0.4 ± 0.2 mJy and a size smaller than 0farcs2. Combined with the broad-band SED of the flare, the radio properties of C1 yield a lower limit of ≈ 0.5 mG for the magnetic field and a total minimum energy of 1.2 × 1041 erg vested in the flare (corresponding to using about 0.2% of the pulsar spin-down power). The 1.6 GHz observations provide upper limits for the brightness (0.2 mJy/beam) and total flux density (0.4 mJy) of the optical Knot 1 located at 0farcs6 from the pulsar. The absolute position of the Crab pulsar is determined, and an estimate of the pulsar proper motion (μα = -13.0 ± 0.2 mas/yr, μδ = + 2.9 ± 0.1 mas/yr) is obtained.