We performed single electron tunneling measurements on bilayer ruthenate Sr$_3$Ru$_2$O$_7$. We observe an unusual oscillation in tunneling magnetoresistance near the metamagnetic quantum phase transition at temperatures below 7 K. The characteristic features of this oscillation suggest that it is unrelated to traditional quantum oscillations caused by orbit quantization. In addition, tunneling spectra are found to change sharply in the low bias voltage range of $V$ $<$ 2mV near the transition field. These observations reveal that the Fermi surface of Sr$_3$Ru$_2$O$_7$ changes in a surprising way as the system undergoes strong critical fluctuations.
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