As outdoor gatherings and sport resumes across England, Matt Hancock said international travel had not been ruled out, though the priority was enabling people in the UK to see friends and family.
He told ITV's This Morning the "biggest problem" was from variants such as those first found in South Africa and Brazil, and it was not yet known if vaccines were effective against them.
"We're not yet sure, but we're doing the science in Porton Down, and watching very closely, and if that all goes well, then we haven't got a problem and then we'll be much more relaxed about international travel," he said.
"We will know more over the next few weeks."
Work is under way at Public Health England's Porton Down laboratory and specialist facilities to find out whether existing vaccines are less protective against variants.
Firms including AstraZeneca, are already working on new jabs that could be rolled out in the autumn if the current crop are found not to work.
Mr Hancock said he had a "lot of sympathy" for the travel industry, adding: "We all want to get that going as soon as possible, but the most important thing is to protect the recovery here at home so people can see their loved ones and don't have to go into lockdown again.
"We may get to a position where if you've had your jab, then other countries will say 'you've got to have the jab to come in'. And so we'll make sure everybody can do that."
Asked if there may be foreign holidays this summer, he said: "There may well be, I wouldn't rule that out.
"The door is not shut, it's just too early to say, but what we can say with confidence is that the unlocking at home is on track."
It comes after scientific experts said summer staycations should be encouraged over foreign holidays this year due to the risk of importing variants of concern.
Dame Anne Johnson, professor of epidemiology at University College London, told BBC Radio 4's Today programme: "This is a risk where you've got high rates of infection. I'm for staycations."
Professor Sir Mark Walport, a former government chief scientific adviser, said data should be one of the deciding factors in whether people can travel abroad.
He said: "Certainly at the moment many countries in Europe have got case numbers that are going up - there is 36,000 cases a day in France, 16,000 in Germany, 22,000 in Italy.
"The numbers speak for themselves."
Overseas holidays are currently banned, but the government's Global Travel Taskforce is due to provide a report to the Prime Minister early next month setting out recommendations for how and when foreign holidays could resume.
Under both Scotland and England's road maps for easing pandemic rules, the earliest date that overseas leisure travel could be permitted is May 17.
Regarding his own summer plans, Mr Hancock told This Morning: "I'm going on holiday in the UK. I can't wait.
"I love this country, I've had my holiday booked since last year."