Around 5.4 million EU citizens have been granted the right to stay in Britain since Brexit. However, several hundred thousand applications are still being processed by authorities.
Mr Foster warned any EU nationals who attempt to remain in the UK illegally will be required to leave.
Speaking to Italian newspaper La Repubblica he said: “EU nationals who have been refused status, and who remain in the country, will be in the UK illegally.
“We expect people to leave the country voluntarily.”
If this doesn’t happen he said immigration officials “will seek to enforce their departure”.
Mr Foster added the UK system has “safeguards” for those who missed the settled statue application deadline, through no fault of their own.
He commented: “Those with reasonable grounds for missing the deadline [to apply to stay in Britain] are still able to apply.
“We have built-in safeguards to protect those who have not yet applied, but who may still be eligible which means everybody will be able to get the status they deserve.”
The Government is expected to announce more work visas will be made available for lorry drivers, as the UK contends with a shortage of around 90,000.
Speaking to the BBC he said: “The market is stretched, so I think that is a broader issue that is affecting the supply chain, not just the petrol and diesel but retail as well.
“The good news is you can only really fill up once; you’ve got to use the fuel, so this should be a short-term thing.
“It’s not like the fuel crises in the past when the supplier was hit by strikes, etc.
“So, once people have filled up, they won’t travel more than they normally travel, so this strain on the system should ease up in the next few days.”
Tensions between London and Brussels are increasing again over Northern Ireland.
Unionists are furious about the Northern Ireland protocol, which puts barriers on trade between the province and other parts of the UK.
Paul Givan, the Northern Irish first minister, is threatening to collapse the Stormont government unless the EU makes concessions.
Lord Frost, a Cabinet Office minister negotiating for the UK, has urged Brussels to help address the situation “urgently”.
He tweeted: “The Protocol is clearly having a continued negative effect on everyday life & business in Northern Ireland.
“The outstanding issues now need to be dealt with urgently.
“I and my team are in contact with the EU daily, but we need a full response to our July Command Paper soon.”