Culture Secretary Lisa Nandy has said she is confident disagreements with campaigners over a new law designed to stop cover-ups can be resolved before it returns to Parliament on Monday.

The Hillsborough Law would introduce a legal obligation for public authorities to cooperate and tell the truth in inquiries.

However, bereaved families and some Labour MPs have raised concerns the heads of security services would be able to exempt MI5 and MI6 officers from disclosing information.

Nandy insisted security services would not be exempt from the legislation but she told the BBC the challenge was also ensuring officers, who often held confidential information, could continue to do their jobs.

The draft law, formally known as the Public Office (Accountability) Bill, would place the same “duty of candour” on security service personnel as other public servants.

However, under a change proposed by the government, this would be subject to the approval of the head of their service.

Campaigners have argued this would allow those running security services to decide whether to disclose information and said they cannot support the bill in its current form.

Elkan Abrahamson, a lawyer for the Hillsborough Law Now campaign, said this would leave them “unchallengeable”.

He said it should be up to the head of an inquiry to decide whether information was relevant, adding there were already national security exemptions that allowed evidence to be heard in private.

Families bereaved by the 2017 Manchester Arena attack have also called for the law to apply fully to security services.

A public inquiry found MI5 had not given an “accurate picture” of the key intelligence it held on the suicide bomber who carried out the attack, which killed 22 people and injured hundreds.

On Saturday the Labour mayors of Greater Manchester and the Liverpool City Region, Andy Burnham and Steve Rotherham said the government’s proposal on the security services “creates to broad an opt-out and risks undermining the spirit of the legislation”.

The government is facing the prospect of a rebellion by some of its own MPs on Monday, when the House of Commons is due to vote on the legislation.

Asked if the government would be able to do a deal with campaigners before a vote in the House of Commons on Monday, Nandy told the BBC’s Sunday with Laura Kuenssberg programme: “I think we’re going to be able to find a way to resolve it. I’m really confident about that.”

She added that the government was listening to families and “we will find a way to get this right because in the end we want the same thing”.

Challenged over whether the government was watering down the legislation, Nandy said: “The security services won’t be exempt but there is a challenge that we’ve got collectively and the challenge is to make sure that this applies to the security services, without fear or favour, that they can continue to do their jobs – they’re often the recipients of confidential information.

“But that we never ever end up in a situation like we did with the Manchester Arena inquiry… where the security services are able to withhold information and present an inaccurate picture to families and to a public inquiry for a very long time.”

The Labour mayor of West Yorkshire, Tracy Brabin said it was “vital” security services were subject to the duty of candour.

She told the programme the government was “95% there” but “without the trust and confidence of the families it’s going to be very difficult”.

On Friday, government sources told the BBC that while they had the the deepest respect for bereaved families, they had gone as far as they could without compromising national security.

Around 30 Labour MPs have backed a proposal by Liverpool Labour MP Ian Byrne, which would ensure the duty of candour applies in full to individuals working for security services.

Byrne – a longstanding campaigner for the law – has said he cannot support it in its current form.

Another Liverpool MP, Anneliese Midgeley has accused the government of breaking its promises to Hillsborough families and urged ministers to change course.

The Hillsborough law follows campaigning by families affected by the 1989 stadium crush in Sheffield, which led to the death of 97 football fans.

Police leaders were found to have spread false narratives about the disaster, blaming Liverpool fans, and withheld evidence of their own failings.