At atmospheric pressure, hydrocarbon resources, such as bitumen and Maya and other heavy oils, have been shown to exhibit non-Newtonian behavior at or below typical ambient temperatures. This work is devoted to investigating the effect of pressure on the non-Newtonian rheological properties of Maya crude oil (a commercial heavy oil blend from Mexico). Measurements were performed from 258 to 333 K, using a stress-controlled rheometer, at pressures up to 150 bar and over a broad range of shear rates. Maya crude oil was shown to be a shear-thinning fluid below 313 K exhibiting thixotropy below 293 K, at atmospheric pressure, in a prior work. At fixed temperature, the magnitude of the non-Newtonian behaviors of Maya crude Oil appears to increase with increasing pressure, and shear thinning is shown to persist to higher pressures below 313 K. Boundaries of the non-Newtonian region with respect to temperature, pressure, and viscosity are identified and discussed. The thixotropic behavior of Maya crude oil is also shown to persist at higher pressure, and the recovery of the moduli at rest appears to be faster at elevated pressures than at atmospheric pressure.