In this paper, we study multi-agent consensus problems under Denial-of-Service (DoS) attacks with data rate constraints. We first consider the leaderless consensus problem and after that we briefly present the analysis of leader-follower consensus. The dynamics of the agents take general forms modeled as homogeneous linear time-invariant systems. In our analysis, we derive lower bounds on the data rate for the multi-agent systems to achieve leaderless and leader-follower consensus in the presence of DoS attacks without quantizer saturation. The main contribution of the paper is the characterization of the trade-off between the tolerable DoS attack levels for leaderless and leader-follower consensus and the required data rates for the quantizers during the communication attempts among the agents. To mitigate the influence of DoS attacks, we employ dynamic quantization with zooming-in and zooming-out capabilities for avoiding quantizer saturation.