In this paper and its companion paper [14], the problem of cooperative source seeking by a formation of mobile agents is considered. Each agent is equipped with position and signal strength measurement sensors; their task is to find the maximum of the scalar field. Agents exchange information with neighboring agents through a communication network. In the first part of this couple of papers, a distributed gradient estimation for each agent and a decentralized navigation controller for single- and double-integrator models are presented. When the signal measurements are corrupted by noise, distributed consensus filters are used in order to estimate the gradient direction. The strategy is based on both a gradient estimating algorithm and a formation controller. Stability conditions are provided. Numerical simulations illustrate the effectiveness of the proposed control law. Part two extends this approach to general linear time-invariant models.