  We investigated the light propagation by means of the Robertson-McVittie
solution which is considered to be the spacetime around the gravitating body
embedded in the FLRW (Friedmann-Lema{\^i}tre-Robertson-Walker) background
metric. We concentrated on the time delay and derived the correction terms with
respect to the Shapiro's formula. To relate with the actual observation and its
reduction process, we also took account of the time transformations; coordinate
time to proper one, and conversely, proper time to coordinate one. We applied
these results to the problem of increase of astronomical unit reported by
Krasinsky and Brumberg (2004). However, we found the influence of the
cosmological expansion on the light propagation does not give an explanation of
observed value, $d{\rm AU}/dt = 15 \pm 4$ [m/century] in the framework of
Robertson-McVittie metric.
