We estimate the amount of Ar37 produced in natural xenon via cosmic-ray-induced spallation, an inevitable consequence of the transportation and storage of xenon on the Earth's surface. We then calculate the resulting Ar37 concentration in a 10-tonne payload (similar to that of the LUX-ZEPLIN experiment) assuming a representative schedule of xenon purification, storage, and delivery to the underground facility. Using the spallation model by Silberberg and Tsao, the sea-level production rate of Ar37 in natural xenon is estimated to be 0.024 atoms/kg/day. Assuming the xenon is successively purified to remove radioactive contaminants in 1-tonne batches at a rate of 1 tonne/month, the average Ar37 activity after 10 tons are purified and transported underground is 0.058-0.090 μBq/kg, depending on the degree of argon removal during above-ground purification. Such cosmogenic Ar37 will appear as a noticeable background in the early science data, while decaying with a 35-day half-life. This newly noticed production mechanism of Ar37 should be considered when planning for future liquid-xenon-based experiments.