We performed a series of resistivity measurements on a MnSi single crystal at high pressures, created by a piston-cylinder device with a liquid pressure medium. The form of the resistivity curve at ambient pressure clearly indicates a first-order nature of the magnetic phase transition in MnSi. Application of high pressure rapidly degrades the first-order features of the phase transition. The temperature derivative of resistivity demonstrates two notable features of the phase transition that disappear on increasing pressure: a sharp peak marking the first-order phase transition and a shallow maximum situated slightly above the critical temperature and pointing to prominent helical fluctuations. The current experimental data rule out any strong first-order phase transition in MnSi at high pressures and low temperatures, which would prevent development of a quantum critical region. On the contrary, there should exist true quantum critical phenomena in MnSi at high pressures because a weak first-order transition, if it survives at high pressures to the lowest temperatures, should not suppress the entire quantum critical region.