The library, Te Matapihi ki te Ao Nui, has been closed since it was deemed an earthquake risk in March 2019.
City councillors have agreed to strengthen the building with base isolators, which is expected to cost $187.4 million.
The Auditor-General's office confirmed it received a letter in December raising concerns about the procurement process.
"After that, in line with our usual process, we considered the issues raised in the letter, and have since been doing some work on those issues," a spokesperson said.
Former National Party MP Chris Finlayson. Photo / Mark Mitchell
The Auditor-General's office does not give public updates on progress or the substance of what it's doing while work is under way, as this can hamper its work, the spokesperson said.
Te Matapihi project board chairman James Robert said the council has been co-operating with the Auditor-General's office since it was made aware of the complaint this year.
"We're looking forward to hearing the findings of the investigation. In the meantime, we're satisfied that our procurement process has been lawful and appropriate."
In March last year the Auditor-General declined to investigate unrelated concerns raised with his office regarding a last-minute proposal to sell part of the central library building.
Labour ticket councillor Fleur Fitzsimons argued recent decision-making over the building wasn't consistent with the Local Government Act or in line with good governance.
Wellington City Council has released artist impressions of the preliminary design of Te Matapihi ki te Ao Nui, the central library. Image / Supplied
But Auditor-General John Ryan replied to Fitzsimons saying he did not plan to investigate the matter further.
He said it was not generally the role of his office to determine if a council had complied with its legal obligations.
The council has recently released artist impressions of the preliminary design for the revamped library.
The renders show better accessibility and connection with the Te Ngākau Civic Square precinct by moving the existing entrance via the stairs to ground level rather than mezzanine level.
A cafe and seating are likely to be in this corner entrance.
Inside, the building is proposed to be much lighter and brighter, with a shaft from the ceiling to bring in natural light.
The design also proposes new entrances from the City to Sea ramp and another on the corner of Harris St. The main entrance on Victoria St will remain.
The design team included Athfield Architects, who designed the building originally; Aurecon, RCP, RLB, Tihei, Art of Fact, and Māpuna.
LT McGuiness have been selected to carry out the construction and plan to prepare to begin earthworks early next year, depending on the resource consent process.
