And further pressure was heaped on Dr Reilly after the family of a pregnant Indian woman, who died in a Galway hospital following a miscarriage, gave him a deadline to announce a public inquiry into her death before they go to the European Court of Human Rights.
"I have every confidence in Dr Reilly," said Mr Kenny. "The investigations for the case arising from the Galway incident are now under way and I hope they will provide some answers to some unanswered questions and in the shortest possible time."
Two probes, by the Health Service Executive and the Health Information and Quality Authority (Hiqa), are being carried out into the death of Savita Halappanavar. However, her husband Praveen refused to co-operate with the investigations, calling instead for a sworn public inquiry.
He has given Dr Reilly until Thursday to announce a public inquiry and has threatened to go to Europe with his demands. His 31-year-old wife died at Galway University Hospital last month following a miscarriage 17 weeks into her pregnancy. She suffered from septicaemia.
Mr Halappanavar claimed she was refused an abortion, despite numerous requests after being told the baby would not survive.
The tragedy, which has sparked pro-choice protests around the globe, reignited an ongoing political debate on whether Ireland should legislate on abortion.
Cabinet ministers will discuss an expert group report this week, which is expected to recommend that new laws on limited abortion be introduced.
But the Taoiseach shot down calls for politicians to be given a free vote in the Dail on the issue of abortion, insisting TDs should vote along party lines.