Mr Varadkar said next Friday would be a reasonable cut-off point to get a deal done ahead of the following week's summit in Brussels, but that an extra 24 or 48 hours could be made available for last-minute talks.
French President Emmanuel Macron urged Boris Johnson to engage in discussions swiftly with the EU's chief negotiator, during a Sunday telephone call in which the British prime minister outlined his latest Brexit proposals, an Elysee official said. If not the United Kingdom will leave with no deal.
Johnson describes the plan as a broad "landing zone" and a spur to further talks.
Meanwhile, Brexit Secretary Stephen Barclay told the European Union and opposition MPs: "We are not backing down".
Ms James told BBC Radio 4's The Week In Westminster she thought she and the other sacked rebels would be able to support the PM's proposals.
But in a snub to David Frost, Boris Johnson's Europe advisor, the European Commission turned down the request, stating that EU member states had agreed the proposals "do not provide a basis for concluding an agreement".
He said they need to get into intense negotiations now with the European Union and what the United Kingdom is clear on and committed to is securing a deal.
- Dutch PM Mark Rutte said he spoke to Mr Johnson on Saturday, but that "important questions remain about the British proposals".
Writing in the Sun on Sunday newspaper, Johnson said the proposals were a "practical compromise that gives ground where necessary while still protecting the UK's interests and the Brexit this country voted for".
But in identical articles for two Brexit-backing British tabloids, Johnson insisted the country will leave the bloc later this month.
Johnson proposed to replace the Irish backstop, one of the stumbling blocks of Brexit talks, with two borders - a UK-EU customs border between Northern Ireland and the Republic of Ireland and a new border between Northern Ireland and the rest of the United Kingdom.
Johnson wants Northern Ireland's devolved assembly - which has been suspended for nearly three years - to vote every four years on whether to maintain European Union rather than British regulations there.
He has also proposed the province leaves the EU's customs union along with the rest of the United Kingdom, with required checks to rely on untried technology and carried out away from the sensitive border.
Describing his blueprint for an agreement as a "practical compromise that gives ground where necessary", Mr Johnson said it represents the United Kingdom "jumping to the island in the middle of the river".
It sees the potential for rampant smuggling while Ireland is concerned hardline Northern Irish unionists would have an effective veto.
Mr Barclay said if Britain does not get a deal in time, the Benn Act will be applied and he confirmed the government will abide by the law.
"The secretary general did not impose an ultimatum but noted that we need clarity before the end of next week if the European Council is to succeed", the official said.
Taoiseach Leo Varadkar, however, has said that while a deal on Brexit is still possible, the current proposals put forward by the United Kingdom do not form the basis for deeper negotiations.