Yet, as speculation began to mount that Mr Johnson might be giving way on customs arrangements to avoid a hard Irish border, the leader of the Democratic Unionist Party (DUP) that is in coalition with the ruling Conservatives issued a statement saying that, "the United Kingdom must leave the European Union as one nation and in so doing that no barriers to trade are erected within the UK".
It noted that any deal must contain a "legally operative solution" that avoids a hard border on the island of Ireland while protecting the island's economy, the Good Friday peace accord and the EU's single market.
Donald Tusk, the European Union council president, said on Friday he had told the prime minister to present his Brexit proposals to the European Union by next Thursday but added that "positive signals" were now emerging from London.
While the details of Johnson's latest proposals have not emerged from the meeting, Varadkar said: "I had a very good meeting today with the prime minister and our teams together - very positive and very promising".
That led to a Brussels working breakfast on Friday between European Union negotiator Michel Barnier and British Brexit minister Stephen Barclay to break the talks logjam.
"Unfortunately we are still in a situation in which the United Kingdom has not come forward with a workable realistic proposal", he added.
The Prime Minister cautioned it was not a "done deal" and there was still "a way to go" if they were to get an agreement which would enable Britain to leave on October 31, as he has promised.
"Of course, there is no guarantee of success and the time is practically up".
Talks would now enter a secretive format of negotiations known as the "tunnel"-used by Johnson's predecessor Theresa May during the final round of negotiations that led to her signing the EU Withdrawal Agreement last November".
Boris Johnson's government is set to enter intensive "tunnel" negotiations with the European Union over a possible Brexit deal ahead of a crunch summit in Brussels next weekend, in a major boost for the prospect of an agreement before the October 31 deadline.
The Taoiseach said a deal was possible after more than two hours of talks but warned that "there's many a slip between cup and lip".
The Prime Minister has so far refused to say if he has backed down over the Irish border issue.
The plan is meant to avoid the need for customs controls between Northern Ireland and the Republic.
Without a deal, Mr Johnson will face demands from opposition parties to comply with the so-called Benn Act which would require him to request a three-month Brexit delay if there is no agreement by October 19.
Whatever the outcome of the "tunnel" talks and next week's summit, United Kingdom lawmakers will be recalled on October 19, a Saturday, for an emergency parliamentary session.
Even if Johnson is able to secure a deal, he would have to win over the DUP, or some or all of the 21 rebels in his own Conservative Party, as well as a section of the 20 or so Leave-supporting Labour MPs opposed to a no-deal Brexit.