In present manuscript, we introduce and study two families $\mathcal{B}_{\Sigma}(\lambda,\delta;\alpha)$ and $\mathcal{B}_{\Sigma}^{*}(\lambda,\delta;\beta)$ of holomorphic and bi-univalent functions which involve the Borel distribution series. We establish upper bounds for the initial Taylor-Maclaurin coefficients $|a_2|$ and $|a_3|$ for functions in each of these families. We also point out special cases and consequences of our results.