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 measure-
ments 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 regularised trans-
formation media theory is employed to arrive at our results. Our proof relies on the study of the long
time behaviour 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 problem. We then present some numerical examples to illustrate our theoretical results.