In a rapidly solidified V41Ni36Si23alloy, the microdomain structure of a pentagonal Frank-Kasper phase (I phase) and the icosahedral as well as the decagonal quasicrystals, giving different tenfold electron diffraction patterns, have been found. Formation of quasicrystals in alloys which would crystallize into Frank–Kasper phases, having a large proportion of icosahedra in their equilibrium structures, is discussed.