Theresa May will come under pressure on Monday to disclose whether any terror suspects are believed to have entered the country when border controls were secretly relaxed this summer.

The home secretary will attempt to defuse the row by telling MPs that she is to set up an inquiry into the UK Border Agency. Union leaders blame cuts and say the relaxation was ordered by ministers.

Hundreds of thousands of people are thought to have entered Britain without being checked against the Home Office index of suspected terrorists and illegal immigrants. The head of the border force, Brodie Clark, has been suspended and an inquiry has been set up under John Vine, chief inspector of the agency.

Conservative MPs will demand assurances that the border is secure, while Labour claims that the minister has presided over a mess at the border force. Shadow home secretary Yvette Cooper said on Sunday that urgent steps were needed to establish whether the public was at risk.

In a letter to May, she said: "The first, and crucial, step must be to ascertain the implications of the lapses in security and passport checks.

"In particular, we need to know whether anyone posing a threat to Britain's national security was allowed to enter the UK during the period where the decision of ministers to relax passport checks was taken further than the Home Office has said was ordered."

Cooper said the public were "understandably appalled and shocked" at the reported failings and urged that Vine's inquiry be "all-encompassing", covering the Home Office, ministerial decision-making and staff cuts.

The Public and Commercial Services Union (PCS) has claimed that border controls were relaxed to keep queues down because of cuts to personnel. It also said the decision was authorised by ministers. May is due to make a statement to the Commons on Monday.

According to reports, border guards were told this summer not to check biometric chips on the passports of citizens from outside the European Union to ensure they are not entering the country fraudulently.

The guards were also instructed not to check fingerprints and other personal details against a Home Office database of terror suspects and illegal immigrants, it was claimed.

Sue Smith, of the PCS, blamed what she said had been a 10% reduction in border force personnel. "The travelling public understandably want to have a fast and efficient service, and yet we are also under a reduced workforce," she said. "So, I think senior managers have seen this as a way to provide the public with what they want."

Senior managers had told the union that the changes had been made with the authorisation of ministers.

"As far as our staff were concerned, this was all done with ministerial authority, and that's the information we have received," she said.

Chris Bryant, the shadow home office minister, called for the publication of all the paperwork between ministers and the agency.

May's team have claimed that the Home Office did ask border staff to streamline their checks on EU nationals entering the country, but primarily because they wanted them to devote more energy to looking at potential illegal immigrants or terrorists.

Although immigration queues were building up at Heathrow airport and other entry points over the summer, May's allies say they were not told they were a major cause for concern.

May was said to have been angered when she found out controls had been loosened. Successive Labour home secretaries – notably John Reid – also complained about their inability to get a grip on the immigration system. He famously described it as "not fit for purpose".