Provided are a computer program product, system, and method to allocate a segment of a buffer to each of a plurality of threads to use for writing data. Each of a plurality of threads are assigned to one of a plurality of segments in a buffer, wherein the threads write to the segment to which they are assigned. A free segment list indicates segments which are not assigned to one of the threads. In response to one of the segments assigned to one of the threads becoming a full segment having less than a threshold amount of free space, indicating the full segment assigned to the thread in the free segment list and assigning one of the segments in the free segment list to the thread different from the full segment.