This paper explores the concept of near-cloaking in the context of time-dependent heat propagation. We show that after the lapse of a certain threshold time, the boundary measurements for the homogeneous heat equation are close to the cloaked heat problem in a certain Sobolev space norm irrespective of the density-conductivity pair in the cloaked region. A regularized transformation media theory is employed to arrive at our results. Our proof relies on the study of the long time behavior of solutions to the parabolic problems with high contrast in density and conductivity coefficients. It further relies on the study of boundary measurement estimates in the presence of small defects in the context of steady conduction problems. We then present some numerical examples to illustrate our theoretical results.