The Ferrari star was speaking during a question-and-answer session when broadcaster David Croft observed that Hamilton appeared happier this season, both on and off the track, prompting loud cheers from fans in attendance.
A fellow racing driver then jokingly asked whether Hamilton's improved mood was down to driving a faster car or having a new girlfriend.
Smiling at the crowd's reaction, Hamilton replied:
“Yeah, that’s a good one.”
Croft also joked about Kardashian's reported attendance at the Monaco Grand Prix, referencing an incident in which she was said to have used a towel reserved for Mercedes rookie Kimi Antonelli.
Hamilton went on to reflect on his first season with Ferrari, describing the experience as both challenging and rewarding as he adapts to life with Formula 1's most successful and iconic team.
“Once you spend a year with this team, Ferrari is still the most iconic team of all time, and they’ve had a difficult period of time,” he said.
“It’s such an amazing team and they’ve been so welcoming. The first year was obviously very tough but to finally see the hard work we’re all putting in, it’s finally getting us back to where we want to be.”
The seven-time Formula 1 world champion then added with a smile:
“And of course, of course, it’s Kim.”
The remark drew loud cheers from the audience and immediately fuelled further discussion on social media, with many fans interpreting it as a rare public reference to Kardashian amid months of dating speculation.
Hamilton and Kardashian have been the subject of persistent media reports linking them romantically, although neither has publicly confirmed the nature of their relationship.
Earlier reports also claimed Kardashian attended the Monaco Grand Prix, where Hamilton was said to have acknowledged her support after the race.
“It’s amazing to have her come this weekend and have the support,” he said at the time.
“But, you know, with my friends, incredible turnout, just overall, the people. It’s amazing to have good people around you and good people supporting you and she does that for me every day.”
Hamilton has historically kept his personal life out of the spotlight, making any public comments about his private relationships a subject of significant interest. His latest remarks at Silverstone are likely to add to ongoing speculation, even as neither he nor Kardashian has officially addressed the reports surrounding them.
Away from his personal life, Hamilton remains focused on helping Ferrari return to championship contention as the Formula 1 season continues.