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. He said: "I wouldn't rule anything out."
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."
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.
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."
The Health Minister said he would reflect carefully on everything he discussed during his private meeting with Mr Halappanavar. But he insisted no decision could be taken regarding a public inquiry until the HSE returns its report. An interim report is expected before Christmas.
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."