The home secretary, Theresa May, on Sunday challenged "scaremongering" claims that London is facing a Romanian crime wave, as she outlined some of the key elements of the government's major immigration bill to be published later this week.

May has made clear that the level of crime by foreign nationals in the capital is directly in line with their representation in the population: "About a third of crime in London is committed by foreign nationals and about a third of the population of London are foreign nationals," she told the BBC's Andrew Marr Show.

Her comments follow a statement by the head of Europol last week that there will be no significant increase in the threat from Romanian criminal gangs when labour market restrictions in Britain on Romanians and Bulgarians are lifted in January. The Ukip leader, Nigel Farage, has repeatedly claimed that Britain will face a Romanian and Bulgarian crime wave from next January.

The home secretary confirmed that a pilot scheme in London, under which immigration officers based in police stations will identify and process foreign suspects on arrest, would now go national.

The officers run immediate checks to see whether the foreign suspect has previous convictions, is wanted abroad or is here illegally. Those who are not prosecuted for the suspected offence may face deportation if they are found to be in Britain illegally or to have a criminal record abroad. They will face exclusion for up to 10 years. May said that more than 1,000 foreign criminals had been removed over the past few months as a result of the pilot scheme.

A new police intelligence assessment to be published on Monday is expected to say that there are 5,500 active organised crime groups in Britain. They are said to include 7,400 "high harm" foreign criminals out of 37,000 gang members.

The home secretary defended her decision to introduce an extended "deport first, appeal later" approach to about half the 64,000 people facing deportation every year, which she announced at the Conservative party conference. She insisted that the process would remain fair: "It is wrong that somebody can stay in the UK, foreign criminals for example, and appeal again and again and build up extra rights to stay here. It is important we allow a fair right of appeal but outside the UK," she said.

May confirmed that the immigration bill being published this week would include curbs on the access of illegal migrants to health, housing and benefit services. She said that among the options being considered to curb "health tourism" was requiring that those who were in Britain only temporarily paid a health levy before they could access the NHS.

A £200 levy has been mentioned as a possible charge facing overseas students from outside the EU. The home secretary said that she also wanted to see the existing arrangements tightened up for recouping the cost of emergency healthcare for foreign nationals from their countries.

May also left open the possibility that her controversial "go home, or face arrest vans" may yet be seen on the streets of Britain again. She refused to rule out their use, saying she was waiting to see a proper evaluation of their impact: "The purpose was to encourage those who were here illegally to go home voluntarily and obviously there is a benefit to government if they do that."

She added: "What we will do is evaluate and once that is done we will take a decision."