And the issue of Northern Ireland's borders could yet break the deal.
The Prime Minister will have a cabinet meeting on Tuesday before she heads to Brussels and she may struggle to get a consensus.
The Government said there were still "unresolved issues" relating to the backstop but that it remained committed to making progress at the European Council meeting.
As for May, she not only has to win over her continental counterparts but also increasingly restive allies back home.
Sunday's flurry of activity comes as British Prime Minister Theresa May prepares to face what one newspaper cartoon dubbed "hell week".
But a spokesman for May said there were "a number of means of achieving what we want to achieve" on the backstop.
She will tell Mr Varadkar that she wants to see a deal that works for both jurisdictions.
Mrs May's counter-proposal, set out in June, was for a "temporary customs arrangement" for the whole United Kingdom, but Tory Brexiteers are suspicious this could turn into a permanent situation - restricting the freedom to strike trade deals around the world.
"I am cautiously optimistic that we can take steps next week but a lot depends on the talks happening in the coming days", Rutte said at his weekly press conference. "That's what we are focused on".
May has insisted she will not accept any deal that creates "a border down the Irish Sea" but recent reports have indicated that her government may be willing to compromise. "This is something which Parliament has already unanimously rejected and is not acceptable to the Prime Minister". "The constitutional and economic consequences of such an approach would be catastrophic in the long run".
Health Secretary Matt Hancock told BBC television that any customs backstop would be "time limited", but did not say whether an expiry date would be written into the deal.
"That is all we are asking for, that's all the Michel Barnier taskforce is also looking for now in terms of legal text".
EU President Donald Tusk has described the summit starting Wednesday as a "moment of truth" in the Brexit talks.
"That needs to be looked at and scrutinised in Parliament". "In all member states, preparation for all eventualities are ramping up quite significantly".
Leaders had been due to decide on Wednesday whether enough progress had been made for them to agree to hold another summit, pencilled in for November 17-18, at which both the treaty on an orderly British withdrawal and a vaguer document setting out future trade relations could be inked in.
The febrile atmosphere in the Tory ranks has seen former Brexit secretary David Davis emerge as a potential successor to Mrs May.
Tory MP Nadine Dorries publicly suggested he could be the leader to deliver the kind of Brexit sought by Eurosceptics.
The warnings issued to Mrs May about the DUP's "red lines" on Brexit are created to signal Mrs Foster's tough leadership qualities to her members - but Nigel Dodds, the party's Commons leader, is increasingly being talked up as her replacement.
If we were to allow this economic annexation of Northern Ireland, by a foreign power, we would be treating Northern Irish MPs as somehow second class legislators, deprived now and forever of any say in numerous laws operational in their own constituencies.