A dislocation in a Dirac semimetal carries an emergent magnetic flux parallel to the dislocation axis. We show that due to the emergent magnetic field the dislocation accommodates a single fermion massless mode of a corresponding low-energy one-particle Hamiltonian. The mode is propagating along the dislocation with its spin directed parallel to the dislocation axis. In agreement with the chiral anomaly observed in Dirac semimetals, an external electric field to the spectral flow of the one-particle Hamiltonian by pumping the fermionic quasiparticles out from vacuum and creating a nonzero axial (chiral) charge in the vicinity of the dislocation. In the presence of the chirality imbalance, the intrinsic magnetic field of the dislocation generates an electric current along the dislocation axis. We point out that this effect – which is an " intrinsic " analogue of the chiral magnetic effect – may experimentally reveal itself through transport measurements in Dirac semimetals via enhanced conductivity when the external electric field is parallel to the dislocation axis.