Raudenbusch and Byrk (2002) write on p. 261:
If an omitted level-2 predictor is associated with a level-I predictor, the coefficient for that level-1 predictor will be estimated with bias. In this case, we have $Cov(u_{qj}, X_{q'ij}) \neq 0$. This problem can be solved, however, without elaborating the level-2 model. Any covariance between a level-1 predictor, $X_{qij}$ and a level-2 random effect must operate through the covariance between the group mean $\bar{X}_{qj}$ and that random effect. Such a covariance can be eliminated by group-mean centering the level-1 predictor.
Alternatively, inclusion of $\bar{X}_{qj}$ as a covariate in each level-2 equation will eliminate any confounding between $X_{qij}$ and omitted variables at level 2.
These two approaches suggest a specification test for the effect of omitted level-2 predictors on a fixed level-1 coefficient. One runs the model with and without group-mean centering of the level-1 coefficient. If the fixed level-1 coefficient associated with that level-1 predictor remains essentially unchanged, the model is not vulnerable to this type of bias.
They don't seem to pass any judgment on which of these two approaches should be preferred if the model is vulnerable to this type of bias. What should drive such a decision? Is one of the methods generally better?
Raudenbush, S. W., & Bryk, A. S. (2002). Hierarchical linear models: Applications and data analysis methods (Vol. 1). sage.