Since the conformal transformations of metric do not change its causal structure, we use these transformations to embed the Lemaître metrics into the FRW background. In our approach, conformal transformation is in agreement with the universe expansion regimes. Indeed, we use the Lemaître metrics because the horizon singularity is eliminated in these metrics. Moreover, some physical and mathematical properties of the introduced metrics have been addressed. We show that the resultant metrics include event horizons while their physical radii are increasing as a function of the universe expansion which may provide suitable metrics for investigating the effects of the universe expansion on the black holes.
