David Cameron’s pledge to reduce annual net migration to below 100,000 during this parliament has begun to publicly unravel as the home secretary, Theresa May, said it was just a “comment” and the prime minister’s spokesman described it as an “objective”.

Ministers admit in private that they will fail to meet the target by the time of the election, and this view burst into the open as May indicated that the cause of the failure would be Britain’s inability to control migration from within the EU.

Speaking on BBC Radio 4’s Today programme, the home secretary made clear that the government was preparing the ground for a public admission of failure on the migration target.

Asked to explain the missed target, May said: “When we made that comment, when we said … we would be aiming to bring the net migration down to the tens of thousands and we wanted to do that within this parliament – yes we were very clear that was what we wanted to do.”

When pressed about whether the target was a promise, commitment, aspiration or comment, Cameron’s official spokesman said: “There is no change. It remains the objective towards which the prime minister and others are working … It has always been the objective.”

The cautious remarks by the home secretary, who stumbled slightly as she referred to the net migration target as “that comment”, contrasted with the unequivocal “no ifs, no buts” declaration made by the prime minister in April 2011.

Cameron highlighted a series of measures to control immigration, adding: “I believe that will mean net migration to this country will be in the order of tens of thousands each year, not the hundreds of thousands every year that we have seen over the last decade.

“Yes, Britain will always be open to the best and brightest from around the world and those fleeing persecution. But with us, our borders will be under control and immigration will be at levels our country can manage. No ifs. No buts. That’s a promise we made to the British people, and it’s a promise we are keeping.”

May said the government had been successful in reducing migration from outside the EU to the lower levels of the 1990s. She said the failure to control migration from within the EU explained why Cameron had placed the free movement of people on the table in his renegotiations of Britain’s EU membership terms.

“The issue is free movement within the EU,” she said. “But we’ve been very clear that this is an issue that we wish to deal with. We think there should be changes in relation to the way free movement operates. David Cameron has been very clear about that and he has said he is going to be speaking more about this before the end of the year.”

Boris Johnson said last month that the prime minister’s net migration target was one of “two big deceptions” on migration, the other being the decision of Tony Blair’s government to open up Britain’s borders 15 years ago. The London mayor told the Andrew Marr Show on BBC1 that Cameron had been wrong in “saying that we could control the numbers when we couldn’t”.

May also launched a strong defence of the government’s decision to opt back in to the European arrest warrant. MPs will vote on government plans to opt back in to 35 EU justice and home affairs measures on 1 December while permanently opting out of about 100 measures. Tory rebels, who had hoped to attract 100 Conservative votes, are now suggesting that their numbers will be reduced to around 30.

May warned that a failure to maintain UK membership of the European arrest warrant could make the UK attractive to criminals evading justice. “If we had a situation where we didn’t have an extradition arrangement with other countries inside the EU and you are a criminal and committed an offence in another country inside the EU, where do you think you might make for if you were trying to evade justice? I think you would probably go to the country where there wasn’t an extradition arrangement. If we don’t have the European arrest warrant, that would be the UK,” she said.