A mould-supporting system for a machine for forming hollow glassware includes a supporting structure, and a first carriage and a second carriage, which are slidably mounted, respectively, on a first bar and a second bar of the supporting structure, and, moreover, are slidably mounted on a third bar of the supporting structure, common to the two carriages, the first and second carriages carrying, respectively, a first mould-carrier arm and a second mould-carrier arm. Mutual displacement of the first and second carriages is controlled between a position where they are set close to one another and a corresponding position where they are set apart. The first and second carriages have a first tubular portion that engages, respectively, the first and second bars via first sliding means, and a second tubular portion that engages the third bar via second sliding means. The second sliding means enable adjustment of the position of the second tubular portion in a direction transverse to the third bar.