We use Chandra X-ray observations to look for evidence of a recoiling black hole from the brightest cluster galaxy in Abell 2261 (A2261-BCG). A2261-BCG is a strong candidate for a recoiling black hole because of its large, flat stellar core, revealed by Hubble Space Telescope imaging observations. We took 100-ksec observations with Chandra and combined it with 35 ksec of archival observations to look for low-level accretion onto a black hole of expected mass $Msim10^{10} M_{scriptscriptstyle odot}$ that could possibly be located in one of four off-center stellar knots near the galaxys center or else in the optical center of the galaxy or in the location of radio emission. We found no X-ray emission arising from a point source in excess of the cluster gas and can place limits on the accretion of any black hole in the central region to a 2-7 keV flux below $4.3 times 10^{-16} mathrm{erg s^{-1} cm^{-2}}$, corresponding to a bolometric Eddington fraction of about $10^{-6}$. Thus there is either no $10^{10} M_{scriptscriptstyle odot}$ black hole in the core of A2261-BCG, or it is accreting at a low level. We also discuss the morphology of the X-ray emitting gas in the cluster and how its asymmetry is consistent with a large dynamic event.
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