In this work, we have studied the $\Lambda_b \rightarrow J/\psi K^{-} p$ decay via $\Lambda^*$-charmonium-proton intermediate states and discussed all possible triangle singularities. Using this process we have done a detailed analysis of the singularities of the triangle amplitude and derived a formula for an easy evaluation of the singularities. We have stressed that the $\chi_{c1}$ and the $\psi(2S)$ are the relatively most relevant states among all possible charmonia up to the $\psi(2S)$. Particularly the $\Lambda(1890)\,\chi_{c1}$ pair plays a very special role, since the threshold and triangle singularities merge. In the case o of $J^{P}=\frac{3}{2}^-,~\frac{5}{2}^+$ for the narrow $P_c$, one needs $P$- and $D$-waves, respectively, in the $\chi_{c1}\, p$. This feature reduces the strength of the contribution and smoothens very much the peak. In this case the singularities cannot account for the observed narrow peak. On the other hand, for the case of
$J^{P}=\frac{1}{2}^+$ or $\frac{3}{2}^+$ with the latter one of the favored quantum numbers, where $\chi_{c1}\, p \rightarrow J/\psi\, p$
can proceed in an $S$-wave,
the $\Lambda(1890)\,\chi_{c1}\,p$ triangle diagram could play an important role.