The head of the European Commission has said that only "miracles" can move Brexit talks far enough forward to fulfill Britain's hopes of launching discussions next month on its future ties with the EU.
Jean Claude Juncker made the comments at an informal summit of EU leaders in Estonia.
It comes after British Prime Minister Theresa May urged the EU to respond in kind to proposals she set out in Italy last week that sought to unlock Brexit talks, saying forming a new partnership was in the bloc's interest as well.
Britain had aimed to make a breakthrough at a summit in Brussels on 19 and 20 October.
Two years have been set aside for the Brexit talks and Britain risks crashing out of the 28-state bloc without a deal on future trade terms come 30 March 2019.
However, Mr Juncker said the first stage of talks on the rights of expatriates, the border between Northern Ireland and the Republic and the financial settlement when Britain leaves had not gone far enough.
He told reporters in Tallinn: "By the end of October, we will not have sufficient progress.
"At the end of this week, I am saying that there will be no sufficient progress from now until October unless miracles will happen."
However, Taoiseach Leo Varadkar has said there is a better mood and better "vibe" coming out of the Brexit negotiations, but it was still "very evident" that a lot more work had to be done.
"We're not yet at a stage where sufficient progress has been made to allow us to talk about the new relationship in trade," said Mr Varadkar, who is attending an informal summit of European Union leaders in the Estonian capital Tallinn.
"I don't think we'll be able to make that call until much later in the month."
The Taoiseach said he would like to see a trade deal done with the four South American countries of Mercosur, "just as we have a very successful one now with Canada," but that standards and Irish farmers needed to be protected.
Ireland is among 11 countries led by France which have written to the European Commission to voice concern about plans to open up farm trade with Brazil, Argentina, Uruguay and Paraguay.
"My view is that free trade makes everyone better off, but it's not an absolute thing, and we do need to take particular care around the beef industry, which may become very exposed depending on the type of Brexit that occurs," said Mr Varadkar.
"I want to make sure there's no diminution of animal health standards, animal welfare standards, and that our farmers can compete on a level playing field."
Mr Varadkar said he believed the United States' threat of huge tariffs against Canadian aerospace company Bombardier - which employs 4,500 people in its Belfast plant - "could well turn out to be a lesson for the United Kingdom".
He said: "When the Brexit referendum went through and in the months since then, there's been a lot of talk about a new trade deal between the UK and the United States and how great that would be for the United Kingdom.
"And yet, we're now talking about the possibility of a trade war between the UK and the United States. I think what it demonstrates is that every country in Europe is actually a small country.
"Britain isn't even in the top 20 countries in the world in terms of population. Germany is 18th. We're actually stronger together as a trading bloc. Perhaps that's something for people to consider."
Taoiseach standing firm against EU taxation moves
Mr Varadkar has rejected moves to change the way digital companies are taxed at EU level, following a push by the big four countries - France, Germany, Spain and Italy - to have major tech companies taxed on turnover rather than where the company is registered.
Arriving at the EU Digital Summit in Tallinn, Mr Varadkar said he had raised the issue with other leaders over an informal dinner last night.
"The point I made was that if we want to foster innovation, if we want Europe to become a digital leader, the solution is not more taxes and more regulation, it's actually the opposite.
"People bemoan the fact that there is no European Google, no European Facebook, there is no European Linked-In.
"My view is that if you want those things in Europe and you want those types of companies to be generated in Europe, it's not through heavy taxes and higher regulation that you're going to achieve that."
The Taoiseach said countries similar to Ireland had a different view compared to the big four especially the Nordic and Benelux companies.