Influenza A is one of the best studied viruses and a model organism for the study of molecular evolution; in particular, much research has focused on detecting natural selection on influenza proteins. Here, we study the dynamics of the synonymous and non-synonymous nucleotide composition of influenza A genes. In several genes, the nucleotide frequencies at synonymous positions drift away from the equilibria predicted from the synonymous substitution matrices. We investigate possible reasons for this unexpected behavior by fitting several regression models. Relaxation towards a mutation-selection equilibrium following a host jump fails to explain the dynamics of the synonymous nucleotide composition, even if we allow for slow temporal changes in the substitution matrix. Instead, we find that deep internal branches of the phylogeny show distinct patterns of nucleotide substitution, and that these branches strongly influence the dynamics of nucleotide composition, suggesting that the observed trends are at least in part a result of natural selection acting on synonymous sites. Moreover, we find that the dynamics of the nucleotide composition at synonymous and non-synonymous sites are highly correlated, providing evidence that even non-synonymous sites can be influenced by selection pressure for nucleotide composition.