We discuss the $pp \to pp \phi$ and $pp \to pp \phi \phi$ reactions
in identifying 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
the photon-pomeron fusion is the dominant process
and the odderon-pomeron fusion is an interesting alternative.
Adding odderon exchange
improves description of the proton-proton angular correlations
and the $dP_{t}$ dependence 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 $\phi\phi$ invariant masses and
large rapidity distance between $\phi$ mesons seems well suited
to distinguish the odderon contribution from other contributions.
Comparisons with exclusive data for single and double $\phi$ production
from the WA102 experiment are made
and predictions for LHC experiments are presented.