Managing multiple locks for data set members in a data set index

Provided are a computer program product, system, and method for managing multiple locks for data set members in a data set index. The data set index has leaf nodes and internal nodes. In response to determining that a current lock is set at a locked node that prevents a grant to a requested lock, a determination is made as to whether there are different first and second descendant nodes from the locked node at a same level of the tree data structure linked to the requested data set member and the locked data set member, respectively. If so, the requested lock for the requested data set member is set at the first descendant node, the current lock is moved to the second descendant node for the locked data set member and the requested lock is granted.