In a meeting with envoys of the bloc's remaining 27 countries on Friday, the EU's chief Brexit negotiator, Michel Barnier, suggested that Johnson is softening his stance on both customs and Stormont's consent.
The prime minister put forward revised proposals for a deal last week, created to avoid a hard border on the island of Ireland after Brexit.
Meanwhile, European Council president Donald Tusk handed the PM a last-minute reprieve to secure an agreement, but warned the United Kingdom still has not presented a "workable, realistic proposal".
The shadow Brexit secretary said that if the prime minister is unable to reach an agreement with Brussels by 31 October, he must comply with the Benn act and seek a further delay.
"There is a joint feeling that there is a way forward, that we can see a pathway to a deal", he said.
"A week ago I told Prime Minister Johnson that if there was no such proposal by today I would announce publicly that there are no more chances because of objective reasons for a deal during the coming European council".
"And Boris Johnson knows it very well".
"If Boris Johnson does manage to negotiate a deal then we will insist that it is put back to the people in a confirmatory vote", he said.
The tepid positivity came after Mr Johnson and Mr Varadkar said they could "see a pathway" to a possible agreement following face-to-face discussions in the Wirral.
Meanwhile, Mr Johnson will challenge MPs to back any deal he has secured from Brussels when parliament sits on a weekend next Saturday for the first time in nearly four decades, The Times has reported. "That kind of deal can never be one Labour supports", he said.
It comes after an apparent Brexit breakthrough on Thursday when the Prime Minister held crunch talks with his Irish counterpart Leo Varadkar.
Barnier's meeting with ambassadors was still going on, but officials with knowledge of the talks said that the 27 other European Union countries had responded positively.
Meanwhile, British and European Union officials are to continue Brexit talks over the weekend amid speculation a deal is possible, which could break the deadlock over the border.
Asked whether the British province of Northern Ireland might remain in the EU customs union after Brexit, Johnson refused to give what he called a "running commentary" on the negotiations.
"No, it can not work", Mr Dodds said, "because Northern Ireland has to remain fully part of the United Kingdom customs union".
Following the meetings on Friday, the European Commission reiterated its position, stating that "there must be a legally operative solution in the Withdrawal Agreement that avoids a hard border on the island of Ireland, protects the all-island economy and the Good Friday (Belfast) Agreement in all its dimensions, and safeguards the integrity of the Single Market".