Negotiation is a fundamental challenge for AI agents, as it requires an ability to reason strategically, model opponents, and balance cooperation with competition. We present the first comprehensive study that systematically evaluates how explicit reasoning training affects the negotiation abilities of both commercial and open-weight large language models, comparing these models to their vanilla counterparts across three languages. Using a self-play setup across three diverse dialogue games, we analyse trade-offs between performance and cost, the language consistency of reasoning processes, and the nature of strategic adaptation exhibited by models. Our findings show that enabling reasoning -- that is, scaling test time compute -- significantly improves negotiation outcomes by enhancing collaboration and helping models overcome task complexities, but comes at a substantial computational cost: reasoning improves GPT-5's performance by 31.4 % while increasing its cost by nearly 400 %. Most critically, we uncover a significant multilingual reasoning distinction: open-weight models consistently switch to English for their internal reasoning steps, even when negotiating in German or Italian (and thus possibly impacting potential explainability gains through the disclosure of reasoning traces), while a leading commercial model maintains language consistency between reasoning and final output.