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 betagamma(= beta/square root(1-beta(2))) region of phi mesons (betagamma < 1.25) with copper targets. However, in the high betagamma region (betagamma > 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 publications on rho/omega modification, this study has experimentally verified vector meson mass modification at normal nuclear density.