  Recently, in their pioneering work on the subject of bi-univalent functions,
Srivastava et al. \cite{HMS-AKM-PG} actually revived the study of the
coefficient problems involving bi-univalent functions. Inspired by the
pioneering work of Srivastava et al. \cite{HMS-AKM-PG}, there has been
triggering interest to study the coefficient problems for the different
subclasses of bi-univalent functions. Motivated largely by Ali et al.
\cite{Ali-Ravi-Ma-Mina-class}, Srivastava et al. \cite{HMS-AKM-PG} and
G\"{u}ney et al. \cite{HOG-GMS-JS-Fib-2018} in this paper, we consider certain
classes of bi-univalent functions related to shell-like curves connected with
Fibonacci numbers to obtain the estimates of second, third Taylor-Maclaurin
coefficients and Fekete - Szeg\"{o} inequalities. Further, certain special
cases are also indicated. Some interesting remarks of the results presented
here are also discussed.
