Referring Expression Segmentation (RES) aims to provide a segmentation mask of the target object in an image referred to by the text (i.e., referring expression). Existing methods require large-scale mask annotations. Moreover, such approaches do not generalize well to unseen/zero-shot scenarios. To address the aforementioned issues, we propose a weakly-supervised bootstrapping architecture for RES with several new algorithmic innovations. To the best of our knowledge, ours is the first approach that considers only a fraction of both mask and box annotations (shown in Fig. 1 and Table 1) for training. To enable principled training of models in such low-annotation settings, improve image-text region-level alignment, and further enhance spatial localization of the target object in the image, we propose Cross-modal Fusion with Attention Consistency module. For automatic pseudo-labeling of unlabeled samples, we introduce a novel Mask Validity Filtering routine based on a spatially aware zero-shot proposal scoring approach. Extensive experiments show that with just 30% annotations, our model SafaRi achieves 59.31 and 48.26 mIoUs as compared to 58.93 and 48.19 mIoUs obtained by the fully-supervised SOTA method SeqTR respectively on RefCOCO+@testA and RefCOCO+testB datasets. SafaRi also outperforms SeqTR by 11.7% (on RefCOCO+testA) and 19.6% (on RefCOCO+testB) in a fully-supervised setting and demonstrates strong generalization capabilities in unseen/zero-shot tasks. Our project page can be found at https://sayannag.github.io/safari_eccv2024/ .