In this paper, we first propose a class of bivariate shrinkage estimators based on Steins unbiased estimate of risk (SURE). Then, we study the effect of correlation coefficients on their performance. Moreover, under some mild assumptions on the model correlations, we set up the optimal asymptotic properties of our estimates when the number of vector means to be estimated grows. Furthermore, we carry out a simulation study to compare how various bivariate competing shrinkage estimators perform and analyze a real data set.