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.