Summary: The main objective of this investigation is to introduce and study the coefficient bound problems for two new subclasses of the familiar class of normalized bi-univalent functions on the open unit disk by using concept of the conic domains. We derive non-sharp bounds on the first two Taylor-Maclaurin coefficients \(|a_2|\) and \(|a_3|\) for functions in each of these general subclasses. We also point out several corollaries and consequences of each of our main results and indicate their connections with some of the earlier known developments.