Let \({\mathcal B}\) denote the set of natural numbers which are a sum of two squares, and let \[ B_ 2(x,k,\ell)=| \{n\leq x: n, n+1\in {\mathcal B},\quad n\equiv \ell (mod k)\}| \quad. \] \textit{G. Bantle} [Math. Z. 189, 561-570 (1985; Zbl 0545.10029)] obtained an upper bound for \(B_ 2(x,k,\ell)-B_ 2(x-x^{\delta},k,\ell)\) for \(k\ll \log^ Ax\) and \(\delta >75/307\). The present authors employ a nice new sieve technique and prove a general result, a corollary of which substantially improves Bantle's result: Let \(\ell \in {\mathbb{N}}\) and \(00\) such that \[ B_ 2(x,k,\ell)-B_ 2(x-x^{\delta_ 1},k,\ell)\leq \frac{c}{k}\prod_{p\equiv 3 (mod 4),\quad p| k}\frac{p}{p-2}\cdot \frac{x^{\delta_ 1}}{\log x}. \]