In this paper we find bounds on the modulii of the second, third and fourth Taylor-Maclaurin's coefficients for functions in a subclass of \emph{bi-close-to-convex} analytic functions, which includes the class studied by Srivastava et al. as particular case. Our estimates on the second and third coefficients improve upon earlier bounds. The result on the fourth coefficient is new. Our bounds are obtained by refining well known estimates for the initial coefficients of the Carthéodory functions.