The growth of biological tissues, which is regulated by a variety of factors, can induce stresses that may, in turn, destabilize the tissues into diverse patterns. In most previous studies, however, tissue growth was usually assumed as a prescribed parameter independent of stresses, limiting our understanding of the mechanobiological morphogenesis of real tissues. In this paper, we propose a theoretical model to investigate the mechanobiological response of soft tissues undergoing stress-modulated growth. Linear stability analysis is first performed to elucidate the surface instability mechanism induced by stress-modulated volumetric growth. We further conduct finite element simulations to validate the theoretical prediction and, particularly, to capture the post-buckling pattern evolution. Our results show that the non-uniform stresses, which evolve with the tissue growth and morphogenesis, exert mechanical feedback on the growth itself, producing up-down asymmetric surface morphologies as observed in, for example, the gyrification of human brains and brain organoids. It is also revealed that large residual stresses are unnecessary to cause mechanobiological instability and subsequent asymmetric patterning, which has long been believed to be driven by sufficiently high stresses. The present work could help us to understand the morphological changes of biological tissues under physiological and pathological conditions.