The transverse single spin asymmetries of $\rho^{0}$ production in semi-inclusive deep inelastic scattering (SIDIS) are recently measured by COMPASS. Among them, the Sivers asymmetry can be described by the convolution of the Sivers function and the unpolarized fragmentation function within the transverse momentum dependent factorization. We perform a phenomenological study and find that the COMPASS data can be well described by the nucleon Sivers functions extracted from previous SIDIS data with pion and kaon productions. Therefore, it provides a universality test of the nucleon Sivers functions within the current experimental precision. Based on the result, we predict the Sivers asymmetry of $\rho^{0}$ production in SIDIS at future experimental facilities, such as the electron-ion colliders. The production of other vector mesons, like $K^{*}$, is also discussed.