He said he was anxious principally that the planned agreement could hurt the Northern Irish peace process in which the British government plays an impartial role. A deal with the DUP's 10 MPs would get her over the 326 MPs she needs.
Mr Kenny warned Theresa May on Sunday that power-sharing in Northern Ireland could be undermined if the DUP propped up her government.
Sir John said any return to violence was "somewhat down the road" but said it was "very important there is a honest broker" in Northern Ireland and only the United Kingdom government could take that role.
Ruth Davidson, leader of the Conservatives in Scotland, sought assurances that LGBT rights would be protected after any deal with the DUP, which opposes same-sex marriage.
Asked about the DUP's demands in any discussions of a deal with the Conservatives, and the Government's stance as an honest broker, he said: "That is the danger for the DUP, because the DUP by doing this are putting themselves in a position where they may have to take responsibility for unpopular actions, where they may suppose themselves to the sort of attack that you're making at the moment, so they are taking a significant risk in going in".
They confirmed their joint commitment to restoring a Northern Ireland Executive as soon as possible and agreed that both countries would continue to engage closely to bring about political stability in Northern Ireland.
Any hard border, he said, would be catastrophic.
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The prime minister may not be present as the talks continue because she is heading to Paris for a meeting with newly-elected President Emmanuel Macron.
Northern Ireland has been without an Assembly and Executive for the past three months, following the collapse of power-sharing over a bitter row between the DUP and Sinn Féin.
Last night, Mr Flanagan said the "heavy lifting" to reach a deal would have to be done by the parties.
Commenting on whether the talks might result in the restoring of power sharing institutions due to the threat of the alternative of direct power controlled from Westminster by a Government involved with the DUP, he said: "Well that might be a positive outcome, I think we should be grown up about this".
"If they do not, the power to make decisions passes to others".
"Like the overwhelming majority across the community in Northern Ireland, I believe a devolved government in Belfast is the best way to address the key decisions which affect people's day to day lives - whether these relate to the economy, security, public services or issues of policing and justice, as well as addressing the legacy of the past".
One of the most pressing issues is the state of the border between Northern Ireland and the Republic of Ireland after Britain leaves the EU.
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