Symmetric or asymmetric positioning of intracellular structures including the nucleus and mitotic spindle steers various biological processes such as cell migration, division, and embryogenesis. In typical animal cells, both a sparse actomyosin meshwork in the cytoplasm and a dense actomyosin cortex underneath the cell membrane participate in the intracellular positioning. However, it remains unclear how these coexisting actomyosin structures regulate the positioning symmetry. To reveal the potential mechanism, we construct an in vitro model composed of cytoplasmic extracts and nucleus-like clusters confined in droplets. Here we find that periodic centripetal actomyosin waves contract from the droplet boundary push clusters to the center in large droplets, while network percolation of bulk actomyosin pulls clusters to the edge in small droplets. An active gel model quantitatively reproduces molecular perturbation experiments, which reveals that the tug-of-war between two distinct actomyosin networks with different maturation time-scales determines the positioning symmetry.