  In this paper we introduce the notion of almost flatness for (stably)
relative bundles on a pair of topological spaces and investigate basic
properties of it. First, we show that almost flatness of topological and smooth
sense are equivalent. This provides a construction of an almost flat stably
relative bundle by using the enlargeability of manifolds. Second, we show the
almost monodromy correspondence, that is, a correspondence between almost flat
(stably) relative bundles and (stably) relative quasi-representations of the
fundamental group.
