The current trial is looking at remedies to curtail Google's dominance in online search, as the recent explosion in generative AI services such as ChatGPT has expanded the market.
Newer AI models search the internet to improve results and reduce hallucination, which has been a problem from developers since chatbots started to become popular.
Last year, OpenAI offered to do a deal with Google which would have integrated Google search results into ChatGPT, according to Mr Turley's testimony.
But he says their offer was rejected.
"We have no partnership with Google today," Mr Turley said, according to Reuters.
OpenAI does however have a partnership with Microsoft, which makes the Bing search engine and Edge browser.
Meanwhile, Google has its own suite of generative AI products such as Gemini - a direct competitor to ChatGPT.
The trial is expected to last three weeks and other big tech companies - including Meta, Amazon and Apple - will be keeping a close eye on it, given they are also facing their own monopoly lawsuits from the DOJ.
Separately, reports in the past week suggest OpenAI is looking into creating its own social network.
The Verge reported the project is "still in early stages," but the company and its boss Sam Altman have privately been asking for feedback on a potential rival to X.
OpenAI boss Sam Altman and X boss Elon Musk used to be business partners until a bitter falling-out over the direction of OpenAI a few years ago.
X has its own AI tool Grok, which it has integrated into the social platform.