The aim of this paper is to study the behaviour of certain sequence of nonlinear Durrmeyer operators $ND_{n}f$ of the form
$$(ND_{n}f)(x)=\int\limits_{0}^{1}K_{n}\left( x,t,f\left( t\right) \right)
dt\,\,\,0\leq x\leq 1,\,\,\,\,\,n\in \mathbb{N},
$$
acting on bounded functions on an interval $\left[ 0,1\right] ,$ where $%
K_{n}\left( x,t,u\right) $ satisfies some suitable assumptions. Here we
estimate the rate of convergence at a point $x$, which is a Lebesgue point
of $f\in L_{1}\left( [0,1]\right) $ be such that $\psi o\left\vert
f\right\vert \in BV\left( [0,1]\right) $, where $\psi o\left\vert
f\right\vert $ denotes the composition of the functions $\psi $ and $%
\left\vert f\right\vert $. The function $\psi :\mathbb{R}_{0}^{+}\rightarrow
\mathbb{R}_{0}^{+}$ is continuous and concave with $\psi (0)=0,$ $\psi (u)>0$
for $u>0$, which appears from the $\left( L-\psi \right) $ Lipschitz
conditions.