During her post-Cabinet press conference, Prime Minister Jacinda Ardern said any tourists that enter New Zealand after 16 March and don't quarantine would be deported.
Dr Clark reiterated Ardern's comments on Morning Report.
"We absolutely are interested in any information on anybody that is looking to flout the arrangements that are in place and we will deport them.
"We make no apology, people who are not going to abide by the way we are doing things are not welcome in New Zealand," he said.
Dr Clark said police were seeking tourists that weren't self-isolating, but wasn't aware if any had been found.
He said testing will continue at the discretion of clinicians throughout the country.
"We will continue to test more people, we've ramped up our testing, we've ramped up our capacity, we've got a lot of spare capacity at this stage but things can change dramatically, so we're looking to make sure we can continue to build the capacity test and we're making sure we're testing more people."
Dr Clark said he's confident New Zealand is in a good position if community transmission does occur here.
"It's one of the fears, we've seen it overseas where there has been community transmission, because we've acted early and decisively we've managed to keep Covid-19 at bay as well as anyone else in the world, but we want to ensure that we are keeping on top of that because we are going to see sporadic cases arriving here."
National's health spokesperson Michael Woodhouse said while the government had landed in the right place with yesterday's announcement - it could be too late.
"The likelihood is that transmission is already occurring, because that has been the experience of other countries who took similar action to us.
"I don't know why New Zealand would have thought that they were going to be any different from other advanced nations like Germany, like the United Kingdom and of course like Italy," he said.
- If you have symptoms of the coronavirus, call the NZ Covid-19 Healthline on 0800 358 5453 (+64 9 358 5453 for international SIMs)
This article was first published on Radio NZ.