Provided are a computer program product, system, and method for copying point-in-time data in a storage to a point-in-time copy data location in advance of destaging data to the storage. A point-in-time copy is created to maintain tracks in a source storage unit as of a point-in-time. A source copy data structure indicates tracks in the source storage unit to copy from the storage to a point-in-time data location. An update to write to a source track is received and a determination is made as to whether the source copy data structure indicates to copy the source track from the storage to the point-in-time data location. The update is written to a cache. A copy operation is initiated to copy the source track from the storage to the point-in-time data location asynchronous before the source track is destaged from the cache to the storage unit.