Cross sections for nuclear reactions at beam energies near and below the spherical Coulomb barrier V c were measured in the very heavy collision systems238U +238U and238U +197Au. The most probable reaction channel with mass transfer is the one-neutron transfer. Its excitation function is understood in terms of Rutherford trajectories together with the quantal process of neutron tunnelling over large distances. In addition, the exchange of up to 15 nucleons is observed down to 0.90 V c . The excitation functions for the multi-nucleon transfer products have much steeper slopes than that for one-neutron transfer, and are steeper for238U +197Au than for238U +238U, suggesting that nuclear contact is established in the associated collisions. The angular distribution for one selected multi-nucleon transfer product,227Th, shows that its formation occurs in more central collisions within contact times shorter than about 10−21 s. There is no evidence for very longlived di-nuclear systems in the these reactions.