Tory MP Mark Reckless accompanied Keith Vaz at Luton Airport, claiming he was concerned that large numbers of Romanians and Bulgarians may come to the UK.
I'm here this morning because my party - the Conservatives - we made a promise that we would cut immigration from hundreds of thousands to tens of thousands a year.
And my concern is that if we do see large numbers coming from Romania and Bulgaria then that could knock us off that target and the promise we made to the electorate.
Romanians landing at Luton Airport were greeted by Home Affairs Select Committee chairman Keith Vaz on the first day of new rules allowing the migrants to work in the UK.
He said he saw no evidence of migrants who have "rushed out and bought tickets" because of the lifting of regulations.
He said that most of the people he met already work in the UK and were "coming back after a holiday".
But he also criticised the government for a "lack of robust estimates" of the number of Romanians and Bulgarians expected to arrive in the UK.
British companies are turning to Romanian workers to fill job vacancies as restrictions on the UK labour market are lifted from today.
Employers are advertising nearly 5,000 positions on Tjobs.ro, a Romanian recruitment website which claims to help 160,000 find work.
Some 4,896 jobs were advertised today as being available in England, along with 80 in Northern Ireland, 24 in Scotland and 20 in Wales.
Among them are 50 nursing positions in southern England, 100 private hire taxi drivers, 10 GPs in Liverpool and 20 carpenters in London.
The Government has done "everything possible" to ensure people come to the UK for the right reasons, the Home Office has said.
Speaking ahead of today's rule change, a Home Office spokesman said, "Hard-working people expect and deserve an immigration system that is fair to British citizens and legitimate migrants and tough on those who abuse the system and flout the law.
"We welcome those that want to come here to work and contribute to the economy, but no EU national has unrestricted access to the UK - they must be working, studying or self-sufficient.
"Across Government, we are working to ensure that our controls on accessing benefits and services, including the NHS and social housing, are amongst the tightest in Europe to protect the UK from abuse."
Romanians and Bulgarians have unrestricted access to the UK labour market from today, despite last-ditch efforts to prevent a feared wave of fresh immigration.
As Prime Minister David Cameron and Home Secretary Theresa May roll out tough new measures designed to tackle concerns that the Government is a soft touch on immigration, temporary curbs imposed in 2005 on citizens of Romania and Bulgaria have been lifted.
Ninety senior Conservatives attempted to block the move in a letter to the Prime Minister, arguing he could invoke a clause in EU law to keep the borders shut.
But ministers have denied such a move would be feasible.
A top Tory warned that a minority of the immigrants from Romania and Bulgaria could escalate existing problems, including pickpocketing and defecating on doorsteps.
Philippa Roe, Conservative leader of Westminster City Council, told BBC Radio 4's The World At One programme: "I know the vast majority of Romanians and Bulgarians planning to come to the UK are planning to work and contribute to society here.
"I think the fear that everybody faces, which is reflected in the media, is those that come to Britain and either fail to find jobs and therefore fall back on our welfare system, or those who deliberately come here to pickpocket and aggressively beg.
"We have seen in the past 18 months, particularly the Roma in central London, causing a massive amount of disruption and low-level crime which has made a very negative impact on our communities.
"It's this minority one is really concerned about but it is this minority that has this really big impact.
"You've only got to wander around Marble Arch at 7.30 in the morning to see the camps.
"We have people walking out of their front door to find people sleeping on their front doorsteps, people defecating on their front doorstep.
"It's extremely unpleasant and it goes with the very aggressive begging and pickpocketing and other sorts of crime in the area which affects both residents and tourists."
An easyJet spokesman has said that the airline has not seen any increase in passenger numbers from Bulgaria or Romania to the UK.
He said: "EasyJet currently only flies to two destinations in Bulgaria and Romania and other airlines provide far more frequent services.
"We haven't seen any evidence of an increase to our passenger numbers from January 2014 on these routes."
A spokeswoman for the Romanian foreign ministry has dismissed claims that the UK could see an "invasion" of migrants and said that some media reports have bordered on racism.
Brandusa Predescu told the BBC: "There isn't going to be an invasion of Romanians ... Not all Romanians, young and old, are going to get on a plane.
"The UK will [not be] and is not the preferred destination of Romanians," she added.
Romanian and Bulgarian migrants bring "knowledge and experience" London businesses "badly need" and fill a skills gap that British youngsters cannot, The London Chamber of Commerce has said.
Chief executive Colin Stanbridge said:
Political rhetoric on migrant workers has stepped up in recent days as the deadline for removal of restrictions approaches. However, the debate often fails to take into account the positive benefits that migration brings to the London economy.
Migrant workers are often highly skilled, bringing with them knowledge and experience that London businesses badly need - now - in order to continue to grow.
Of course the Government needs to equip our young people with the skills that businesses require, but that will take years.
Experts predict 250,000 Romanian and Bulgarian migrants will move the UK over the next five years when restrictions on entry to the UK are lifted on January 1.
According to a report by Migration Watch UK:
- A Romanian with two children and a spouse on the average wage in their home country will take home £4,000 after tax
- A Bulgarian in the same circumstance will take home £3,800
- Even if they were still on half the average Romanian or Bulgarian wage, that worker will be six times better off in the UK
- They are three times better off in Spain or Italy
- Germany is the "most attractive destination" as it has a low unemployment rate and large number of vacancies