The frequency-response masking (FRM) approach was introduced as a means for reducing the arithmetic complexity of FIR filters with narrow transition bands. Compared to conventional direct-form realizations, FRM filters may have a substantially lower complexity. However, up to now, only linear-phase FIR filters have been considered. Such filters have long delays which may be unacceptable in certain applications. In this paper, we introduce FRM FIR filters with short delay. The overall filter has approximately linear phase, and its delay can be chosen arbitrarily. The paper considers how to design the filters, and includes design examples demonstrating the benefits of the proposed filters.