Work permits are among the post-Brexit migration curbs being considered but any changes must be good for the UK economy, the home secretary has said.
Amber Rudd told the BBC the work permit proposal "certainly has value" but nothing was being ruled out.
She accepted EU nations could choose to impose new restrictions, including requiring Britons to apply for permission before travelling.
Ms Rudd said it was a "given" people voted Leave to reduce immigration.
Speaking on the Andrew Marr Show, she reiterated the prime minister's dismissal of a points-based system to control EU migration championed by Brexit campaigners in the build-up to the referendum, saying it "simply doesn't work".
She said her department was considering the alternative of requiring EU migrants to have work permits.
"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.
Asked about a possible trade-off between curbing the free movement of people and being allowed access to the EU's single market, Ms Rudd said the UK will have "complete control" over numbers once it leaves the EU, "with one or two provisos".
The new model "has got to be reciprocal", she said.
"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."
- Since the Brexit vote, much of the debate has been about how the government will try to reduce migration from the EU
- During the referendum campaign, Vote Leave called for a "points-based" system, similar to that used in Australia
- This would involve applications being accepted on the basis of their skills, with the criteria and overall numbers to be determined by MPs
- Theresa May has rejected this, saying it would not give sufficient control to the government
- An alternative option would be to require migrants to have a work permit before coming to work in the UK, with ministers able to prioritise different sectors
- A combination of different models is also an option, and the government says all possibilities are being considered
- It has also been reported a visa waiver scheme, similar to that used by the US, could apply to Britons going to the EU
- This could involve an online application and paying a fee in order to visit the EU without requiring a full visa
Asked about the Conservatives' long-standing target to reduce net migration to the tens of thousands - latest figures had it at 327,000 - she said the government was "completely committed" to reducing immigration, and "yes tens of thousands, although it will take some time".
She also confirmed the government was looking at ways of reducing the number of people using student visas to come to the UK, but said there would be "no blanket ban".
The home secretary said reports that Britons may have to pay for permission to visit the EU after Brexit were "a reminder that this is a two-way negotiation" adding: "I don't think it is particularly desirable but we do not rule it out because we have to be given a free hand."
Shadow home secretary Andy Burnham said such a proposal could cost a British family visiting the EU £50.
"This is yet another example of the drift and confusion as a result of the government's failure to plan for Brexit," he added.
Ms Rudd, one of the leading voices in the Remain campaign ahead of the referendum, played down her attacks on Leave campaigner Boris Johnson, who is now foreign secretary.
Asked about her comment in a TV debate that Mr Johnson was "the life and soul of the party" but "not the man you want driving you home at the end of the evening", she said: "Boris is not the the driver, Theresa May is the driver.
"She is very clear that we are all focused in the same direction and we are all going to deliver on what she has asked us to do."