Vector autoregressive (VAR) models aim to capture linear temporal interdependencies amongst multiple time series. They have been widely used in macroeconomics and financial econometrics and more recently have found novel applications in functional genomics and neuroscience. These applications have also accentuated the need to investigate the behavior of the VAR model in a high-dimensional regime, which provides novel insights into the role of temporal dependence for regularized estimates of the model's parameters. However, hardly anything is known regarding properties of the posterior distribution for Bayesian VAR models in such regimes. In this work, we consider a VAR model with two prior choices for the autoregressive coefficient matrix: a non-hierarchical matrix-normal prior and a hierarchical prior, which corresponds to an scale mixture of normals. We establish posterior consistency for both these priors under standard regularity assumptions, when the dimension of the VAR model grows with the sample size (but still remains smaller than ). A special case corresponds to a shrinkage prior that introduces (group) sparsity in the columns of the model coefficient matrices. The performance of the model estimates are illustrated on synthetic and real macroeconomic data sets.