Mr Wolff said Brussels has a "will" to reach an agreement with the United Kingdom, especially in regards to the so-called Brexit bill.
Barnier that he was unable "in the current circumstances" to ask the European Council for permission to begin progressing talks and discuss the UK's future relationship with the EU.
At the centre of the deadlock - a phrase United Kingdom officials were reluctant to use - is that the European Union wants to see guarantees on what the United Kingdom is willing to pay, while the United Kingdom wants the European Union to move first by beginning talks on the future before London reveals its hand on the money issue.
He said the interpretation of rights must be "genuinely consistent" between the United Kingdom and the EU after Britain quits and suggested they must be applied through the European Court of Justice.
Mr. Barnier made clear, however, that in spite of new momentum from concessions given by May in a speech at Florence on September 22, the British positions on money, expatriate citizens' rights and the Irish border still fell short of the "significant progress".
Speaking to the BBC, he added: "The EU has a big interest in finding some deal and getting the United Kingdom to pay especially for 2019-20 where the money is certainly needed".
While that is unlikely to happen, Barnier has been pushing member states for making a gesture towards the United Kingdom by pledging to "scope" the positions on transition within the EU-27.
However, he also acknowledged "new momentum" in negotiations, adding that "decisive progress is in our grasp within the next two months".
The 27 leaders without Britain will declare that there is not "sufficient progress" on divorce issues such as the exit bill to formally move onto the next phase, including a post-Brexit trade deal, according to a draft statement obtained by AFP Thursday. We confined ourselves to technical discussions.
The issue of citizens' rights is one of the main issues to be dealt with during the negotiations, together with the land borders between Northern Ireland and the Republic of Ireland as well as Gibraltar and Spain, plus immigration control and access to the European single market.
On the latter issue differences still remain over the European Court of Justice's role in guaranteeing those rights, family unifications and the exporting of United Kingdom benefits.