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Access NI provides basic, standard and enhanced checks.
Depending on the level of check, different criminal records and cautions are disclosed on a person's certificate.
An enhanced background check would hold information about any ongoing investigation.
According to Mr Gamble, the BHF recruitment process had decided that an enhanced check was not required for the role McMonagle was applying for.
The charity's chief executive Fearghal McKinney previously told BBC News NI "the BHF is confident we undertook all necessary statutory and moral duties required throughout this entire process".
"For you to be subjected to an enhanced check you need to be applying for a role with children and vulnerable adults or in a position of trust," Mr Gamble told the Good Morning Ulster on Wednesday.
"It's about how frequently you will be left alone with children and young people, that will determine whether or not you need this check, and this is the hurdle you have to jump."
Mr Gamble believes anyone that is in a critical role, that is communicating by, for and on behalf of an organisation, representing them, and engaging on their behalf, that role needs to be automatically included for an enhanced check.
Sinn Féin president Mary-Lou McDonald has apologised for party's handling of the the investigation.
In a statement on Monday, she confirmed she has ordered "a complete overhaul of governance procedures" in Sinn Féin.
McDonald said she was disgusted by the crimes of McMonagle and said the decision of two former Sinn Féin staff members to give him a reference was "unforgivable and absolutely unacceptable".
McDonald added she will ensure "clear processes" are in place so anyone suspended or who ends their employment with Sinn Féin will have no "future issues in relation to handing back passes and to make absolutely clear that references cannot be provided by individual members of staff".
Mr Gamble said any investigation into Sinn Féin needs to be an "external independent review with no holes barred so that people can see that you're not just committed to saying the right thing but actually doing the right things".
He said that an independent review needs to make a judgement on whether for some people in the party, protecting the reputation of the organisation was more important than protecting vulnerable young people.
The Department of Justice has been asked for comment.
