The inquiry will investigate the Assembly's two past nominations for Senate casual vacancies, including Senator Gallagher's nomination, as part of a territory response to the widening citizenship crisis engulfing the federal parliament.

It follows an Opposition push for twin inquiries - one into the administrative process and a second "privileges" probe into Senator Gallagher's nomination.

Greens leader Shane Rattenbury, who holds the balance of power in the Assembly, would not back a specific probe into Senator Gallagher's nomination.

All parties in the Assembly have backed an amendment he tabled on Thursday for a wider inquiry into the nominations process, the scope of which would take in Senator Gallagher's nomination.

But Chief Minister Andrew Barr said the inquiry would not have any practical bearing on Senator Gallagher's position, given it did not relate to her contemporary appointment, but an earlier federal parliamentary term.

While Mrs Dunne said it was only "tangentially attached" to Senator Gallagher's declaration, she said the inquiry would investigate the senator's declaration, as the inquiry terms were both prospective and retrospective in nature.

The Assembly has only filled two such casual vacancies since self-government, Senator Gallagher's nomination in 2015 and former Liberal Senator Gary Humphries' in 2003.

A copy of the amendment shows the inquiry would be largely administrative in nature, focussing on the Assembly's process for nominations, given the constitutional crisis facing the federal parliament.

It will also consider such processes in other jurisdictions, whether the two nominations may be considered "in hindsight to be unsound" and what improvements could be made to the Assembly's process.

Senator Gallagher, a former ACT chief minister, renounced any entitlement to British citizenship just before the 2016 election, acting on advice from Labor officials as she nominated for re-election.

But it is unclear whether she was eligible to fill a casual vacancy from March 2015 until she was elected in her own right in 2016, as her father was a British citizen.

Earlier this month, former Home Office lawyer Phillip Gamble said she likely would have acquired British nationality by operation of law at her birth.

Senator Gallagher has refused to say when moves to renounce entitlement to UK dual citizenship were officially confirmed by British authorities.

But she has said she was advised that submitting a renunciation ahead of the 2016 election, on April 20 last year, meant she had taken "all reasonable steps to renounce any entitlement to British citizenship".

The inquiry will report back to the Assembly in March next year.