According to reports in The Sunday Times, the employee in question was hired through a recruitment agency, which failed to carry out the required background checks.
John Barker, managing director at manufacturer of background checking tools TSSI, said: "It is worrying that this is still happening even after the Baroness Scotland incident recently. As a government department this is a poor example to set other UK businesses.
"It is imperative that these pre-employment checks are carried out or agencies risk damaging their reputation or having to pay hefty fines.
"There are tools available that allow employers to check the authenticity of ID documents such as passports, driving licences and Visa documents, ensuring their employees can work in the UK.
"Civil penalties are no idle threat. The UK Borders Authority is cracking down on illegal employment and companies could be looking at fines of up to £10,000 per illegal worker. This will make a huge dent in profits and is not something that businesses, in particular, recruitment agencies that are experiencing huge cut backs, can risk in the current economic climate."
