In this paper we report the electric current and conductivity dependences of ferroelectric layer thickness were studied in a LaCaMnO3/BaTiO3 (LCMO/BTO) heterostructures. Structures LCMO/BTO with respective layer thicknesses of LCMO 12 nm and variable thicknesses of BTO layers were fabricated by aerosol deposition method (MAD) onto MgO (100) substrates. The thickness variations of ferroelectric layers were 2, 4 and 6 nanometers respectively. For all samples two ranges of applied electric field reveal different behavior of electrical parameters. For high fields, the current-voltage dependence shows ohmic behavior. For low field, the current-voltage dependence shows nonlinear behavior accompanied by conductance dependence with hysteresis. This kind of exponential growth indicates the tunneling current through thin ferroelectric layer, which can be modeled by Sommerfeld-Bethe method. The main questions we addressed were: 1) what is the origin of transport mechanism through BTO junction and LCMO layer? 2) how does the thickness of BTO influence on the tunneling through the ferroelectric barrier? 3) how does the tunneling features manifests itself in the current-voltage characteristics? 4) how the interface BTO/LCMO influence on transport mechanism and tunneling through BTO junction. For the heterostructures with 2 nanometers thickness of BTO layers the nonlinear behavior of current-voltage dependence more much evidenced compare with the samples 4 nm and 6 nm thickness of BTO layer. We attribute this fact that the tunneling mechanism directly depends on the ferroelectric layer thickness. Below the metal-insulator transition temperature TMI the BTO layer is dielectric, while the LCMO layer becomes metallic. The ratio between the electric field components in the BTO and LCMO layers is determined by the boundary conditions for tangential field components: EBTO=ELCMO. Then, in the case of an ideal metallic layer, the system may be treated as two ferroelectric capacitors connected in parallel, with the electric field oriented perpendicular to the film plane (z-orientation). Conductivity as well as current in the multilayer with the smallest BTO layer show increasing with increasing voltage. In other both samples we also observed the increasing conductance with increase voltage, but rather smaller. Although in the samples with 4 nm and 6 nm thickness of the BTO layers the nonlinearity in electric current and conductivity dependences were not so pronounced, the switching of resistivity was found in that structures. We suppose that the observed effect appears due to the polarization formation in the ferroelectric films and the electric dipoles oriented along the electric field. After changes direction of electric field also observed change the polarization, but slowly up to coercive force. Thus we observed the hysteresis in conductance dependence.