There is provided a process for producing molten glass, which is capable of easily increasing the H2O content in glass melt with excessive generation of convention of the glass melt being reduced.One mode of the process for producing molten glass according to the present invention is characterized to include a material melting step for melting a raw glass material in a melting furnace to prepare glass melt; a water-molecules supply step for supplying a water-molecules supply gas into the glass melt flowing from an upstream end of the melting furnace toward a downstream end of the melting furnace; and a refining step for degassing, under a reduced pressure atmosphere, the glass melt flowing out of the downstream end; wherein a position where the water-molecules supply gas is supplied in the water-molecules supply step includes a first position and a second position from downstream to upstream in a flowing direction of the glass melt in this order; the first position is a position away from both of the upstream end and the downstream end; and the second position is a position closer to the upstream end than a center of a distance between the upstream end and the first position in the flowing direction of the glass melt.