Button is out of contract at the end of 2014, so many interpreted Dennis' comments as the start of negotiations.
Champion trio Fernando Alonso, Sebastian Vettel and Lewis Hamilton have all been linked with Button's seat, and the 34-year-old Briton - the most experienced driver on the grid today - reacted coolly to Dennis' quip.
"It isn't the only team in formula one," Button said.
It is believed members of McLaren's senior management also did not appreciate Dennis' message, but Neale insists the newspapers the words were printed on should be thrown away.
"It's chip paper," he said at Silverstone.
"I think he (Dennis) did the same thing to Ayrton Senna. I'm pretty sure he did the same thing to Kevin (Magnussen). I think if you listen to my phone on a daily basis he'll be doing the same thing to me," said Neale.
In truth, both Button and rookie Dane Magnussen might have reason to worry.
McLaren is on the hunt for a top driver to lead the new Honda era, and the team ousted newcomer Sergio Perez after a solid single season last year.
Is Magnussen worried?
"Everyone knows at McLaren that if you don't deliver you don't deserve the place," the 21-year-old told F1's official website. "I think that's the way it should be."