Ms Rudd said reports the European Commission is considering plans for a visa waiver programme to operate across the Schengen free movement area is a “reminder” that the UK is in a “two-way negotiation” with the EU as it seeks its divorce from Brussels.
She agreed that people would be “surprised” if they had to apply for short-term leave to visit countries like France but insisted such a scheme could be rolled out.
Labour warned a family of five could face a £50 charge for a European holiday unless the government fought to keep free movement between Britain and the UK in Brexit negotiations.
Last week it emerged that the French and German governments had suggested modelling a European Commission proposal for a common visa waiver programme on the US Esta scheme. Visits to America from countries that do not require full visas must apply online for permission to enter the country, at a cost of $14, or £10.
The Home Secretary told the BBC’s Andrew Marr Show: “I don’t think it’s particularly desirable but we don’t rule it out because we have to be allowed a free hand to give the best negotiation.”
Ms Rudd said the UK will be able to control its borders post-Brexit but stressed any measures introduced would have to be “reciprocal”.
She said: “Once we leave the European Union we will have complete control over who comes into the UK from the EU and who doesn’t, with one or two provisos of course.
“First of all, it’s going to be reciprocal, we are going to have to work out what’s in the UK’s interests as well going to the European Union and what works for our economy and making sure that we get the right balance.
“Looking across the whole spectrum is what’s going to be the guiding principle.”
Ms Rudd also reaffirmed the Tory manifesto commitment to cut immigration to the UK to the tens of thousands, despite not naming any specific measures to achieve a goal that has eluded the government since it was set. “I’m completely committed to making sure that we reduce it and yes, tens of thousands, although it will take some time,” she said.
The Home Secretary said it is “too early” to outline the specifics of how the government will achieve its target but she repeated the Prime Minister’s rejection of an Australian-style points system. However, she suggested a work permit system is being considered.
“Whether we look at a work permit system or another system is something that my department is looking at closely at the moment,” she said.
She said a work permit system “certainly has value” and “we are not ruling anything out at the moment”.
Ms Rudd also responded to concerns that the number of student visas available to people coming to the UK from non-EU countries could be reduced in order to cut immigration numbers.
She said: “We are looking at a number of options but students do make an important contribution.
“There’s going to be no blanket banning of students coming to the UK but we are looking at bringing down the numbers overall.”
Stephen Gethins MP, the SNP’s Europe spokesman, said the UK government was putting the Scottish tourism sector at risk.
“This admission from the Home Secretary that people travelling from the UK to the EU may require a visa will shock many and exposes the serious threat that Brexit poses to an important freedom we have long taken for granted,” Mr Gethins said.
“It also betrays, once again, the Tories’ complete lack of a plan when it comes to Brexit. The ability to travel, live, work and study freely anywhere in the EU is something that we have all got used to and taken advantage of in recent decades.
“It is a huge bonus for Scotland’s economy and culture and was no doubt one of the many positives of EU membership that led to the people of Scotland voting overwhelmingly to remain in Europe.
“Visa-free travel also offers a range of benefits to businesses across the UK but with Brexit Secretary David Davis suggesting that the UK would be unable to stay in the single market – the largest trading bloc in the world – and International Trade Secretary Liam Fox saying UK business executives are ‘too lazy and fat’ to go after exports, it is clear that the Tories have no idea how best to promote and support our businesses overseas.
“Theresa May must now confirm whether these comments are simply Amber Rudd’s personal opinion or if they do reflect government policy.”
Andy Burnham, Labour’s shadow home secretary, hit out at the idea of Britons facing a visa system to visit EU countries.
He said: “This is yet another example of the drift and confusion as a result of the government’s failure to plan for Brexit. Ministers should not just accept there’s a cost of £50 for the average family to go on holiday.
“The Home Secretary’s words will not have reassured ordinary families about the cost of Brexit. She seems to be sympathetic to an idea that will put a flat £50 tax on the average family holiday in Europe.
“Tory ministers might think nothing of that, but it would make it even harder for ordinary families to afford a holiday. Norway doesn’t have the charge so why should we? I challenge the Home Secretary to rule it out.”
Liberal Democrat leader Tim Farron said: “Having to pay for a visa to visit or work in Europe is another sign that things are happening around this Brexit Tory government and all they seem to do is squabble amongst themselves.
“No 10 are not in the driving seat, they are locked in the boot of a process that is threatening to career out of control.”
Ms Rudd addressed comments during the referendum campaign in which she suggested Boris Johnson was not the kind of person you want to drive you home at the end of an evening out.
Following the suggestion that Mr Johnson was now driving Brexit in his role as Foreign Secretary, Ms Rudd said: “Boris is not the driver. Theresa May is the driver. The rest of us are in the car.”