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Statistical bias correction techniques are commonly used in climate model projections to reduce systematic biases. Among the several bias correction techniques, univariate linear bias correction (e.g., quantile mapping) is the most popular, given its simplicity. Univariate linear bias correction can accurately reproduce the observed mean of a given climate variable. However, when performed separately on multiple variables, it does not yield the observed multivariate cross‐correlation structure. In the current study, we consider the intrinsic properties of two candidate univariate linear bias‐correction approaches (simple linear regression and asynchronous regression) in estimating the observed cross‐correlation between precipitation and temperature. Two linear regression models are applied separately on both the observed and the projected variables. The analytical solution suggests that two candidate approaches simply reproduce the cross‐correlation from the general circulation models (GCMs) in the bias‐corrected data set because of their linearity. Our study adopts two frameworks, based on the Fisher z‐transformation and bootstrapping, to provide 95% lower and upper confidence limits (referred as the permissible bound) for the GCM cross‐correlation. Beyond the permissible bound, raw/bias‐corrected GCM cross‐correlation significantly differs from those observed. Two frameworks are applied on three GCMs from the CMIP5 multimodel ensemble over the coterminous United States. We found that (a) the univariate linear techniques fail to reproduce the observed cross‐correlation in the bias‐corrected data set over 90% (30–50%) of the grid points where the multivariate skewness coefficient values are substantial (small) and statistically significant (statistically insignificant) from zero; (b) the performance of the univariate linear techniques under bootstrapping (Fisher z‐transformation) remains uniform (non‐uniform) across climate regions, months, and GCMs; (c) grid points, where the observed cross‐correlation is statistically significant, witness a failure fraction of around 0.2 (0.8) under the Fisher z‐transformation (bootstrapping). The importance of reproducing cross‐correlations is also discussed along with an enquiry into the multivariate approaches that can potentially address the bias in yielding cross‐correlations. [ABSTRACT FROM AUTHOR]