We consider a class of multiscale parabolic problems with diffusion coefficients oscillating in space at a possibly small scale ε . Numerical homogenization methods are popular for such problems, because they capture efficiently the asymptotic behavior as ε→0ε→0, without using a dramatically fine spatial discretization at the scale of the fast oscillations. However, it is known that such homogenization schemes are in general not accurate for both the highly oscillatory regime ε→0ε→0 and the non-oscillatory regime ε∼1ε∼1. In this paper, we introduce an Asymptotic Preserving method based on an exact micro–macro decomposition of the solution, which remains consistent for both regimes.