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 $γ$ and a cyclic $(n,p,n,\frac{n-γ}{p}+γ,0,\frac{n-γ}{p})$-DPDS for an arbitrary integer $γ$. Next, we present the necessary conditions for the existence of a $p$-ary NPS of type $γ$. We apply this result for excluding the existence of some $p$-ary NPS of period $n$ and type $γ$ for $n \leq 100$ and $\vert γ\vert \leq 2$. We also prove the similar results for an almost $p$-ary NPS of type $γ$. 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.