  Gravitational lensing by clusters of galaxies affects the cosmic X-ray
background (XRB) by altering the observed density and flux distribution of
background X-ray sources. At faint detection flux thresholds, the resolved
X-ray sources appear brighter and diluted, while the unresolved component of
the XRB appears dimmer and more anisotropic, due to lensing. The diffuse X-ray
intensity in the outer halos of clusters might be lower than the sky-averaged
XRB, after the subtraction of resolved sources. Detection of the lensing signal
with a wide-field X-ray telescope could probe the mass distribution of a
cluster out to its virialization boundary. In particular, we show that the
lensing signature imprinted on the resolved component of the XRB by the cluster
A1689, should be difficult but possible to detect out to 8' at the 2-4 sigma
level, after 10^6 seconds of observation with the forthcoming AXAF satellite.
The lensing signal is fairly insensitive to the lens redshift in the range
0.1<z<0.6. The amplitude of the lensing signal is however sensitive to the
faint end slope of the number-flux relation for unresolved X-ray sources, and
can thus help constrain models of the XRB. A search for X-ray arcs or arclets
could identify the fraction of all faint sources which originate from extended
emission of distant galaxies. The probability for a 3 sigma detection of an
arclet which is stretched by a factor of about 3 after a 10^6 seconds
observation of A1689 with AXAF, is roughly comparable to the fraction of all
background X-ray sources that have an intrinsic size of order 1''.
