According to People, the Grammy winner kept it low-key, turning down photos with fans as she and Horton exited the restaurant.
However, another source told the publication that Gaga didn't mind being seen with the monitor engineer.
"Ironically, she had a table in the front near the sidewalk, so she was clearly okay with being seen. She seemed very happy when interacting with the guy she was with," said the diner.
Or perhaps the singer didn't realise she could be seen from the balcony, as another source said the two didn't engage in any PDA while ordering their drinks.
Horton has worked alongside the singer since November 2018, before her Las Vegas residency.
The Nashville-based engineer has a penchant for performers and was previously married to Broadway star Autumn Guzzardi.
According to his LinkedIn page, Horton owns a company called Audio Engineering Consulting Group and has previously worked with other stars such as Camila Cabello, Bruno Mars and Justin Timberlake.
Gaga and her ex-boyfriend, Christian, ended their engagement right before the Grammy Awards.
After the 44-year-old actor split with Irina Shayk in June, the rumours only intensified.
However, an entertainment source told People that fans seemed to be jumping to conclusions.
“He has a huge and overwhelming connection to Gaga but whether it becomes a real love story in their lives for all the world to see is premature."