Provided are a computer program product, system, and method for determining whether to extend a drain time to copy data blocks from a first storage to a second storage. A data structure is generated indicating data blocks in the first storage to copy to the second storage. A drain operation is initiated to copy the data blocks indicated in the first storage to the second storage for a drain time period. Write requests to the data blocks indicated in the data structure are queued during the drain time period, wherein the queued write requests are not completed while queued. Metric information based on the writes that occur to data blocks in the first storage are gathered during the drain time period; and in response to expiration of the drain time period, a determination is made from the gathered metric information of whether to continue the drain operation or terminate the drain operation.