Log-structured merge tree (LSM-tree) based key-value stores are widely employed in large-scale storage systems. In the compaction of the key-value store, SSTables are merged with overlapping key ranges and sorted for data queries. This, however, incurs write amplification and thus degrades system performance, especially under update-intensive workloads. Current optimization focuses mostly on the reduction of the overload of compaction in the host, but rarely makes full use of computation in the device. To address these issues, we propose Co-KV, a Collaborative Key-Value store between the host and a near-data processing ( i.e., NDP) model based SSD to improve compaction. Co-KV offers three benefits: (1) reducing write amplification by a compaction offloading scheme between host and device; (2) relieving the overload of compaction in the host and leveraging computation in the SSD based on the NDP model; and (3) improving the performance of LSM-tree based key-value stores under update-intensive workloads. Extensive db_bench experiment show that Co-KV largely achieves a 2.0x overall throughput improvement, and a write amplification reduction by up to 36.0% over the state-of-the-art LevelDB. Under YCSB workloads, Co-KV increases the throughput by 1.7x - 2.4x while decreases the write amplification and average latency by up to 30.0% and 43.0%, respectively.