In an interview with host Emma Barnett, Mr Daisley said, "If someone tweets private information ... private address, phone number -- then there are no-go areas where we don't permit that."
"Were he (Trump) to do that, just picking a hypothetical example, then those would be areas" that were grounds for discipline, the Twitter executive added.
"We would caution him to remove that tweet for sure," he said.
The micro-blogging platform has clarified that it will not block world leaders from tweeting.
Although it didn't directly name Trump, in a blog post called "World Leaders on Twitter", the company said that there has been a lot of discussion about political figures and world leaders on the platform.
"Blocking a world leader from Twitter or removing their controversial tweets would hide important information people should be able to see and debate," the company said.
Twitter did not block Trump for his "nuclear button" tweet that stormed the social media which, many thought, raised prospect of nuclear war with North Korea.
A number of users reported to the tweet, with the expectation that threatening a war one is capable of starting should violate Twitter's revised "Terms of Service", given the company's recent crackdown on violent threats.
The company said it reviews tweets by leaders within the political context that defines them, and enforces its rules accordingly.
"No one person's account drives Twitter's growth, or influences these decisions. We work hard to remain unbiased with the public interest in mind," it added.