The Weibull distribution is one of the most popular distributions in analyzing the lifetime data. In this study, we consider the Bayes estimators of the scale and shape parameters of  Weibull distribution under the assumptions of gamma priors and squared error loss function. While computing the Bayes estimates for a Weibull distribution, the continuous conjugate joint prior distribution of the shape and scale parameters does not exist and the closed form expressions of the Bayes estimators cannot be obtained. In this study first we will consider the Bayesian inference of the scale parameter under the assumption that the shape parameter is known. We will assume that the scale parameter has a gamma prior. Under these assumptions Bayes estimate can be obtained in explicit form. When both the parameters are unknown, the Bayes estimates cannot be obtained in closed form. In this case, we will assume that the scale parameter has the gamma prior, and the shape parameter also has the gamma prior and they are independently distributed. We will use the Lindley approximation to obtain the approximate Bayes estimators. Under these assumptions, we will compute approximate Bayes estimators and compare with the maximum likelihood estimators by Monte Carlo simulations.