Invariant mass spectra of e+e- pairs have been measured in 12 GeV p+A reactions to detect possible in-medium modification of vector mesons. Copper and carbon targets are used to study the nuclear-size dependence of e+e- invariant mass distributions. A significant excess on the low-mass side of the phi meson peak is observed in the low beta gamma (=beta/sqrt(1-beta^2)) region of phi mesons (beta gamma < 1.25) with copper targets. However, in the high beta gamma region (beta gamma > 1.25), spectral shapes of phi mesons are well described by the Breit-Wigner shape when experimental effects are considered. Thus, in addition to our earlier publication on rho / omega modification, this study has experimentally verified vector meson mass modification at normal nuclear density.