There is wide agreement that type Ia supernovae (used as standard candles for cosmology) are associated with the thermonuclear explosions of white dwarf stars. The nuclear runaway that leads to the explosion could start in a white dwarf gradually accumulating matter from a companion star until it reaches the Chandrasekhar limit, or could be triggered by the merger of two white dwarfs in a compact binary system. The X-ray signatures of these two possible paths are very different. Whereas no strong electromagnetic emission is expected in the merger scenario until shortly before the supernova, the white dwarf accreting material from the normal star becomes a source of copious X-rays for about 10(7) years before the explosion. This offers a means of determining which path dominates. Here we report that the observed X-ray flux from six nearby elliptical galaxies and galaxy bulges is a factor of approximately 30-50 less than predicted in the accretion scenario, based upon an estimate of the supernova rate from their K-band luminosities. We conclude that no more than about five per cent of type Ia supernovae in early-type galaxies can be produced by white dwarfs in accreting binary systems, unless their progenitors are much younger than the bulk of the stellar population in these galaxies, or explosions of sub-Chandrasekhar white dwarfs make a significant contribution to the supernova rate.