We have analyzed Einstein High Resolution Imager (HRI) and Imaging Pro
portional Counter (IPC) observations of the gravitationally lensed qua
sar 0957 + 561. The X-ray emission appears as an approximate semicircl
e extending from the northern quasar image (A) to the southern image (
B). We find unresolved X-ray emission associated with B, while A lies
in an extended region. Although, for the epoch of the X-ray observatio
ns, the ratio of the optical flux from image B to that from image A is
0.76, the HRI observations show that the emission from around quasar
image B is only about one-third that from image A. This difference in
ratios implies the presence of another component in 0957 + 561. Since
this ratio is found in each of the two HRI observations which were tak
en 6 months apart, it is unlikely that the difference in the ratios ob
served in X-ray and optical wavelengths is caused by dramatic changes
in the quasar flux on a time scale less than the difference in the lig
ht propagation time for the two images. We use a model for the lens, i
ncluding a giant galaxy and its surrounding cluster, to infer the X-ra
y source distribution and luminosity. This model implies the existence
of an extended source of X-ray emission offset from the unresolved so
urce associated with the quasar nucleus. The extended component has a
diameter approximately 1'' to 3'', offset 0.''7 northeast of the quasa
r. The IPC observation of 0957 + 561 shows that the spectrum of the em
ission is soft compared to that of other X-ray emitting quasars. We di
scuss possible sources for the origin of the X-ray emission and sugges
t that it could arise from radiative cooling of hot gas surrounding th
e quasar. We also suggest that the detection of the extended emission
in 0957 + 561 is possible only because of the large magnification prov
ided by the intervening lens.