The Irish Government would accept unification as way to avoid a hard border, but its preferred solution is the backstop, the Taoiseach has said.
Mr Varadkar said the UK reversing the Brexit decision, or deciding to stay in the single market and customs union, were other potential ways to resolve the border issue.
He said a fifth theoretical solution, Ireland re-joining the UK, was never going to happen.
“There are one in five ways that this can be done, and at least four of those would be acceptable to the Irish Government, but the best one is of course a backstop or some form of backstop, and that’s what we are trying to achieve,” the Taoiseach said on a visit to Sweden.
On the prospect of a united Ireland, Mr Varadkar said: “There probably isn’t a majority for that at the moment, although some opinion polls would suggest there is.”
Commenting on the likelihood of the UK staying in the EU, the Taoiseach added: “All the polls since Prime Minister Johnson became prime minister suggest that’s what the British people actually want, but their political system isn’t able to give them that choice.”
Later, Ireland’s deputy premier said the island of Ireland was not ready for a poll on unification.
Simon Coveney warned that such a debate would create more divisions at a time when the focus should be on the post-Brexit arrangements.
Mr Coveney, addressing the parliamentary Foreign Affairs Committee in Dublin, said it was not the time for a border poll.
“But all of the divisions that flow from that, and the calls for more radical change and so on in Northern Ireland, which I don’t believe we’re ready for,” he said. ”
“So these are difficult times where we’re writing history at the moment for what the relationship between Britain and Ireland and the EU and the UK will look like for the next 20 years and we need to get it right.”
He added: “Ultimately, we’ve got to make sure that we aren’t going back to a border on the island of Ireland being a political debating point again, and all that flows from that in terms of the corrosive impact on relationships and the disruption to trade, which has been such a reinforcer of normality and peace in the border region for the last two decades.
“We want to work with the British Prime Minister and we’re working as part of Michel Barnier’s team.
“We will do everything we can to try to get a deal In the next week or so.
“But it’s got to be the right deal so that we don’t let people down and find that the border becomes the topic of debate in Ireland for the foreseeable future.”
Mr Coveney added: “If we’re going to replace the backstop with something else, it has to do the same job, which is to reassure border communities that they’re not going to face border infrastructure and the disruption in the future.”