As a second investigation into the tragedy was announced - a statutory review by the Health Information and Quality Authority (Hiqa) - Health Minister Dr James Reilly was considering continued requests for an open inquiry.
Mr Gilmore said the priority was getting to the bottom of the 31-year-old dentist's death after a miscarriage.
"I wouldn't rule anything out," Mr Gilmore said.
Ms Halappanavar's husband Praveen has made repeated calls for an inquiry to be public after refusing to co-operate with the two other probes.
After meeting Dr Reilly privately in Galway, Mr Halappanavar said he had been assured that the minister would examine his wishes.
"He just expressed his condolences to the family and I'm going to pass it on to my family and Savita's family back home," he said.
"I'd like to thank him for his condolences and for taking the time to come and meet us."
Dr Reilly met the grieving widower more than a week after the story broke and following repeated suggestions for a Government leader to make a private approach.
Mr Halappanavar is considering taking a case to the European Court of Human Rights over his demands for a sworn public inquiry.
He wants witnesses to be cross-examined and full disclosure of all documentation and communication between medical staff involved in his wife's care and refusal to allow her an abortion.
Another of the serious allegations is that Ms Halappanavar's medical notes include no reference to her multiple requests for a termination.
Staff at Galway University Hospital did note there were requests for tea, toast and a blanket.
Hiqa's statutory inquiry was launched following an approach from the Health Service Executive (HSE) amid concerns over the independence of its own, original inquiry. It is now classed a clinical review.
Mr Halappanavar said he had no confidence in the executive, saying there was a danger it would be biased in favour of staff at the hospital where his wife died.
But the Tanaiste 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."
Mrs Halappanavar died on October 28. She miscarried 17 weeks into her pregnancy and suffered septicaemia.
Her husband claims she made repeated calls for an abortion after learning her baby would not survive, but she was refused the procedure because a foetal heartbeat was present.
He said a consultant later told her: "This is a Catholic country."
The Irish Council for Civil Liberties (ICCL) supports Mr Halappanavar's call for a public inquiry and has expressed doubt that a Hiqa investigation would meet requirements under human rights law.
Hiqa is investigating the safety, quality and standards of services at the Galway hospital towards critically ill patients, including critically ill pregnant women.
The health watchdog said it will publish the terms of reference of its probe and name the investigating team when the details are finalised.
ICCL chief Mark Kelly said that article 2 of the European Convention on Human Rights - cited by Mr Halappanavar's solicitor Gerard O'Donnell as the basis for a public inquiry - entitles the widower to his demands under a number of legal criteria.
These include the need to establish the cause of death of a patient in the care of the medical profession - both in the public and private sector; identifying any responsibility of doctors or other medical staff concerned and holding them to account; and taking all reasonable steps to secure evidence of the incident - through witness testimonies, forensic evidence, post-mortem reports and an objective analysis of all clinical findings.
Other criteria demand an investigation into the death of a patient be carried out promptly and expeditiously; that it includes a certain amount of public scrutiny; and that the next of kin of the dead person is involved to make sure his interests are met.
Mr Kelly said that, while he respected the vigour and professionalism of Hiqa's work, he had concerns about whether an inquiry would satisfy these requirements.
"If this is not the case, Mr Praveen Halappanavar will remain fully entitled to a further inquiry that does effectively discharge the state's responsibilities under the European Convention," he said.
Hiqa said it could not comment on suggestions by Mr Kelly that its investigation would lack transparency as it was now bound by legal restrictions.
Elsewhere, trade union Siptu has heaped further pressure on the Government to legislate on the 20-year-old Supreme Court ruling of the X case.
It called for a change in Irish law allowing a woman an abortion when her life is at risk.
Cabinet members will consider an expert group report on a European Court of Human Rights ruling on the so-called ABC case next week.
The report will be published immediately after Cabinet and a Dail debate on the issue is expected in the coming weeks.
Galway and Roscommon University Hospitals Group has confirmed it will co-operate fully with the Hiqa inquiry.
In a statement, the group confirmed it was already engaged with the HSE investigation and the Coroner.
"As we await the outcomes, we will ensure continuity of the highest levels of care for our patients," it added.
"Galway and Roscommon University Hospitals Group has privately and publicly expressed its condolences to Mr Praveen Halappanavar on the tragic death of his wife, Savita, and would like to take this opportunity to again extend heartfelt sympathy to Mr Halappanavar and his family and friends."