Provided are a computer program product, system, and method for asynchronous pausing of the formation of consistency groups. A first copy operation is initiated to copy the source data from the first storage to the first data copy in the second storage. A second copy operation is initiated to copy the first data copy in the second storage to a second data copy in response to forming a consistency group of the source data at the first data copy forms consistent as of a point-in-time with respect to the source data. The first copy operation is suspended to allow for further processing of the first data copy in response to completing the second copy operation. The formation of a next consistency group is restarted between the source data and the first data copy in response to receiving a resume command.