The EU wants progress to be made before a summit later this month - only then will they agree for trade talks on Brexit to move forward. Officials said he was preparing to call round European Union leaders to get agreement on trade negotiations.
Meanwhile, Sinn Fein's Brexit spokesman David Cullinane told Sky News it is "crucial" Northern Ireland remains in the customs union and single market as "the only way to avoid the hardening of the border".
She said: "There are a couple of issues, some differences do remain which require further negotiation and consultation".
But the issues of the rights of expatriate citizens and the UK-EU border on the island of Ireland remain fraught, diplomats said.
Juncker described May as a "tough negotiator", and insisted that talks had not failed.
"Not every single question has to be answered but we need sufficient progress on these very sensitive issues".
"We now have a common understanding on most related issues with just two or three open for discussion".
Irish state broadcaster RTE said that a draft text would commit Britain to "continued regulatory alignment" with the European Union to avoid divergence on the two sides of the Irish border.
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If she had ignored their concerns, there's little doubt that the party's 10 MPs would have sat on their hands and not supported the Conservatives in important votes on the EU Withdrawal Bill this week.
May depends in parliament on a pro-British party in Northern Ireland that rejects any deal which would divide the province from the British mainland.
"Crucially it is clear that we want to move forward together, but on a couple of issues, differences do remain, which require further negotiation and consultation", said May before heading to meet President of the European Council Donald Tusk nearly one-hour late than scheduled. "The economic and constitutional integrity of the United Kingdom must not be compromised in any way".
"And the problem between the Republic and Northern Ireland is a outcome of the British decision not only to leave the EU but also the single market and the customs union".
He added: "I still hope that this matter can be concluded in the coming days as agreed".
And Verhofstadt warned that unless all issues are solved "there will be no green light in October 2018".
A decision on any new deals with Britain would have to be reached by the fall of next year to give individual member states enough time to approve all the measures in their parliaments before the final date on March 2019.
Last Friday, the European Council President Donald Tusk visited Dublin and told British politicians that, whether they liked it or not, the key to the UK's future after Brexit lay in Dublin.