English: The structure of low-spin states originating from shape-coexisting configurations in 6840Ni28 was directly probed via the two-neutron transfer reaction 66Ni(t,p)68Ni in inverse kinematics using a radioactive ion beam on a radioactive target. The direct feeding to the first excited 0+ state was measured for center-of-mass angles 4∘–16∘ and amounts to an integral of 4.2(16)% relative to the ground state. The observed difference in feeding of the 0+ states is explained by the transfer of neutrons, mainly in the pf shell below N=40 for the ground state, and across N=40 in the g9/2 orbital for the 0+2, based on second-order distorted-wave Born approximation calculations combined with state-of-the-art shell-model two-nucleon amplitudes. However, the direct feeding to the 2+1 state [29(3)%] is incompatible with these calculations.