The degree to which the level of genetic variation for gene expression is shared across multiple tissues has important implications for research investigating the role of expression on the etiology of complex human traits and diseases. In the last few years, several studies have been published reporting the extent of overlap in expression quantitative trait loci (eQTL) identified in multiple tissues or cell types. Although these studies provide important information on the regulatory control of genes across tissues, their limited power means that they can typically only explain a small proportion of genetic variation for gene expression. Here, using expression data from monozygotic twins (MZ), we investigate the genetic control of gene expression in lymphoblastoid cell lines (LCL) and whole blood (WB). We estimate the genetic correlation that represents the combined effects of all causal loci across the whole genome and is a measure of the level of common genetic control of gene expression between the two RNA sources. Our results show that, when averaged across the genome, mean levels of genetic correlation for gene expression in LCL and WB samples are close to zero. We support our results with evidence from gene expression in an independent sample of LCL, T-cells, and fibroblasts. In addition, we provide evidence that housekeeping genes, which maintain basic cellular functions, are more likely to have high genetic correlations between the RNA sources than non-housekeeping genes, implying a relationship between the transcript function and the degree to which a gene has tissue-specific genetic regulatory control.