Federated learning (FL) is a recently developed area of machine learning, in which the private data of a large number of distributed clients is used to develop a global model under the coordination of a central server without explicitly exposing the data. The standard FL strategy has a number of significant bottlenecks including large communication requirements and high impact on the clients' resources. Several strategies have been described in the literature trying to address these issues. In this paper, a novel scheme based on the notion of "model growing" is proposed. Initially, the server deploys a small model of low complexity, which is trained to capture the data complexity during the initial set of rounds. When the performance of such a model saturates, the server switches to a larger model with the help of function-preserving transformations. The model complexity increases as more data is processed by the clients, and the overall process continues until the desired performance is achieved. Therefore, the most complex model is broadcast only at the final stage in our approach resulting in substantial reduction in communication cost and client computational requirements. The proposed approach is tested extensively on three standard benchmarks and is shown to achieve substantial reduction in communication and client computation while achieving comparable accuracy when compared to the current most effective strategies.