The possibility to use the exclusive $pp \to pp \phi$ and $pp \to pp \phi \phi$ reactions in identifying the odderon exchange is discussed. So far there is no unambiguous experimental evidence for the odderon, the charge conjugation $C = -1$ counterpart of the $C = +1$ pomeron. Recently proposed tensor-pomeron and vector-odderon model for soft high-energy reactions is applied. For the $p p \to p p \phi$ reaction at high energies the photon-pomeron fusion is the dominant process and the odderon-pomeron fusion is an interesting alternative. Adding odderon exchange improves considerably description of the proton-proton angular correlations measured by the WA102 collaboration. The $p p \to p p \phi \phi$ process via pomeron-pomeron fusion is advantageous one as here the odderon does not couple to protons. The observation of large $M_{\phi \phi}$ and the rapidity difference $Y_{\phi \phi}$ seems well suited to identify odderon exchange. Comparisons with data from the WA102 experiment are made and predictions for LHC experiments are given.