Provided are a computer program product, system, and method for managing access requests from a host to tracks in storage. A cursor is set to point to a track in a range of tracks established for sequential accesses. Cache resources are accessed for the cache for tracks in the range of tracks in advance of processing access requests to the range of tracks. Indication is received of a subset of tracks in the range of tracks for subsequent access transactions and a determination is made whether the cursor points to a track in the subset of tracks. The cursor is set to point to a track in the subset of tracks and cache resources are accessed for tracks in the subset of tracks for anticipation of access transactions to tracks in the subset of tracks.