  This paper exhibits a Monte Carlo study on site percolation using the
Newmann-Ziff algorithm in distorted square and simple cubic lattices where each
site is allowed to be directly linked with any other site if the euclidean
separation between the pair is at most a certain distance d, called the
connection threshold. Distorted lattices are formed from regular lattices by a
random but controlled dislocation of the sites with the help of a parameter
{\alpha}, called the distortion parameter. The distinctive feature of this
study is the relaxation of the restriction of forming bonds with only the
nearest neighbors. Owing to this flexibility and the intricate interplay
between the two parameters {\alpha} and d, the site percolation threshold may
either increase or decrease with distortion. The dependence of the percolation
threshold on the average degree of a site has been explored to show that the
obtained results are consistent with those on percolation in regular lattices
with extended neighborhood and continuum percolation.
