Let $\mathbb{D}$ be the open unit disk with its boundary $\partial\mathbb{D}$ in the complex plane $\mathbb{C}$ and $dA(z)=\frac{1}{\pi}dx\, dy,$ the normalized area measure on $\mathbb{D}.$ Let $L_{a}^{2}(\mathbb{D}, dA)$ be the Bergman space consisting of analytic functions on $\mathbb{D}$ that are also in $L^2(\mathbb{D}, dA).$ In this paper we obtain certain distance estimates for bounded linear operators defined on the Bergman space.