We present 12CO(J=1-0) and 12CO(J=2-1) spectra for 19 bright, late-type galaxies (spirals) in the central region of the galaxy cluster Abell 1367 (z = 0.02) from observations made with the IRAM 30-m telescope. All 19 spirals were observed at the position of their optical centre and for a subset, at multiple positions. For each spiral the integrated CO (J = 1-0) intensity from the central pointing, in fewcases supplemented with intensities from offset pointings,was used to estimate its molecular hydrogen mass and H2 deficiency. Accepting the considerable uncertainties involved in determining H2 deficiencies, spirals previously identified by us to have redder colours and higher HI deficiencies as a result of environmental influence were found to be more H2 deficient compared to members of the sample in less advanced evolutionary states. For eight of the observed spirals multiple pointing observations were made to investigate the distribution of their molecular gas. For these spirals we fitted Gaussians to the CO intensities projected in a line across the galaxy. In two cases, CGCG 097-079 and CGCG 097-102(N), the offset between the CO and optical intensity maxima was significantly larger than the pointing uncertainty, and the full width at half-maximum values of the fits were significantly greater than those of the other spirals, irrespective of optical size. Both signatures are indicators of an abnormal molecular gas distribution. In the case of CGCG 097-079, which is considered an archetype for ram-pressure stripping, our observations indicate that the CO intensity maximum lies ~15.6 ± 8.5 arcsec (6 kpc) north-west of the optical centre at the same projected position as the HI intensity maximum. ©2012 The Authors Published by Oxford University Press on behalf of the Royal Astronomical Society.