The dynamical evolution of trans-Neptunian objects (TNOs) to the inner Solar system is investigated. The study is based on the observed sample of high-eccentricity TNOs with perihelia in the near-Neptune region, using a procedure to take account of observational biases. It is shown that observations favour TNOs in high-eccentricity orbits as the main source of Jupiter-family (JF) comets. The relative fraction of objects captured per year from the near-Neptune region to JF comets with perihelion distances q < 1.5 au is estimated as 0.2 × 10-10. The maximum lifetime of typical JF comets with q < 1.5 au is approximately 200 revolutions. Based on the observed population of JF comets, there should be ~1010 TNOs of cometary size in high-eccentricity orbits with 28 < q < 35.5 au. If this population originated 4.5 Gyr ago, the primordial number must have been at least 20 times as large as the present one.