A sequence of period n is called a nearly perfect sequence of type gamma if all out-of-phase autocorrelation coefficients are a constant gamma. In this paper we study nearly perfect sequences (NPS) via their connection to direct product difference sets (DPDS). We prove the connection between a p-ary NPS of period n and type gamma and a cyclic (n,p,n, n-gamma/p + gamma, 0, n-gamma/p)-DPDS for an arbitrary integer gamma. Next, we present the necessary conditions for the existence of a p-ary NPS of type gamma. We apply this result for excluding the existence of some p-ary NPS of period n and type gamma for n <= 100 and vertical bar gamma vertical bar <= 2. We also prove the similar results for an almost p-ary NPS of type gamma. Finally, we show the non-existence of some almost p-ary perfect sequences by showing the non-existence of equivalent cyclic relative difference sets by using the notion of multipliers.