We report systematic field-cooled (FC) magnetisation of superconducting (V$_{0.6}$Ti$_{0.4}$)-Y alloys in the presence of applied magnetic fields up to 7~T. A paramagnetic response is clearly observed just below the superconducting transition temperature ($T_{c}$) in low fields ($\leq 0.2$~T). The lower $T_{c}$ of the Y-rich precipitates, as compared to the bulk, is the origin of flux compression that leads to this paramagnetic response in low fields. It is also observed that the magnetisation obtained during field-cooled (FC) cooling cycle is lower than that of FC warming for the (V$_{0.6}$Ti$_{0.4}$)-Y alloys in fields higher than 0.02~T. Moreover, paramagnetic relaxation of the FC moment is also observed in this system. We infer that these features of the Y containing alloys are related to high field paramagnetic Meissner effect (HFPME). Our analysis shows that the large difference in pinning strength of the different pinning centres generated due to Y addition in the V$_{0.6}$Ti$_{0.4}$ alloys, is responsible for the HFPME. We find additional evidence of our claim in the form of extension of range of the temperature and magnetic fields over which the HFPME is observed, when the samples are subjected to cold work. The observation of HFPME up to 7~T has not been reported till now in any low $T_{c}$ superconductor, and only a very few high $T_{c}$ superconductors show HFPME upto such high fields.