We present Very Large Array H i imaging data for a field in the NW of the galaxy cluster Abell 1367 (z= 0.02) in an attempt to probe the effect environment has on the interstellar medium of late-type spiral galaxies. Several galaxies, like CGCG 097-087, show pronounced tails and asymmetries, and seven out of 10 show significant, several kpc offsets between the H i centroid and the optical. We compare our results against a sample of optically bright, late-type galaxies (spirals) across the central 1.5 Mpc of the cluster taken from the Arecibo Galaxy Environment Survey. We find that these late-type spirals are predominantly found in the northern half of the cluster, especially those that are relatively gas rich. We calculate the H i deficiency and find that the expected global trend for the H i deficiency of these spirals to increase with projected proximity to the cluster core, seen in clusters like Coma and Virgo, is not observed. We classified the spirals into four evolutionary states, with the galaxies in each state sharing a similar degree of H i deficiency and optical colour. The common characteristics of the spirals in each evolutionary state suggest they have been subject to similar environmental processes. Many of the spirals in the most common evolutionary state (moderate H i deficiency and blue colour) have an H i intensity maximum which is displaced relative to its optical counterpart. The orientation of these offsets and magnitude of their H i deficiencies together with data from other wavelengths provide observational evidence in support of varying degrees of ram pressure stripping and tidal interaction. In general, our results indicate that the H i discs of bright late-type galaxies in the central part of the cluster are subject to both gas loss and morphological disturbance as a result of their interaction with the cluster environment. This provides further observational evidence of a more complex environment in Abell 1367 as compared to Virgo and Coma. © 2010 The Authors. Journal compilation © 2010 RAS.