"I've made it very clear this investigation must be completed as expeditiously as possible and I now understand that will happen before Christmas and I may have an interim report in a matter of weeks," James Reilly, the health minister, was quoted by the Irish Times as saying.
"I equally understand now that the Health Information and Quality Authority (HIQA) have become involved and they will do that as expeditiously as possible but until I get the report I can't say what the next step is. The report will certainly inform our next actions," he added.
Savita, 31, died from blood poisoning at the Galway University Hospital on October 28 after doctors allegedly refused to perform an abortion stating "this is a Catholic country".
Reilly made his comments at the Mid-Western Regional Hospital in Limerick on Saturday, having earlier met Halappanavar's husband Praveen in a Galway city centre hotel.
Halappanavar said afterwards he was pleased to have finally met a government representative four weeks after his wife died at University Hospital Galway.
But he stressed to Reilly that he does not believe the Health Service Executive (HSE) or HIQA are far-reaching enough.
"I'm just glad that we met and he just passed on his condolences to the family," said Halappanavar, who was accompanied by his solicitor Gerard O'Donnell at the meeting.
"It is (important) for the family because there is no comment or officials who called us or met us to pass on their condolences. So I will pass this on to Savita's family."
"We also said why we need a public inquiry as well and he said he would look into it," Halappanavar said.
Meanwhile, Halappanavar's solicitor O'Donnell said he had taken instructions from his client to seek direction from the Ombudsman on whether he or Galway University Hospital owns her medical records.
Halappanavar has objected to the use of his wife's notes in a HSE inquiry into her death. He has said he has no faith in a HSE-run inquiry and does not want her notes used in it.
O'Donnell had asked that the hospital, where Halappanavar died last month, hand over the original medical notes. However, the HSE has said it owns them.
A spokesman for the Ombudsman has said it was unlikely the office would have a role in this dispute.
Earlier, Eamon Gilmore, the second most senior officer in the Irish government, warned about a tribunal-style inquiry, saying: "We have experience in this country of formal public inquiries and the danger is they go on for a very long time and very often spend a long time being mired in legal argument."
The Irish Council for Civil Liberties has supported Halappanavar's call for a public inquiry.
The Health Information and Quality Authority will publish the terms of reference of its inquiry into her death next week.
The investigation, for which no time span is indicated, will make use of outside expertise, a spokesman indicated.
UN special rapporteur Margaret Sekeggya was in Ireland this week. Outlining that the preliminary findings from her visit she described Ireland's law on terminations as "one of the most restrictive in Europe".