The British had hoped to enter a new, intensive and secretive phase of negotiations known in the Brussels jargon as the tunnel. While European officials said that is not imminent, they did allow that discussions would be accelerated.

Few doubt that the talks are entering a critical phase. A European Union official said that the intensification translated in practical terms into more discussions at the technical level beginning late Friday and going through the weekend. The hope is that the negotiators can reach a breakthrough agreement to present to member states and the European Parliament on Monday, or perhaps earlier.

There are risks, however, that once the details emerge, the talks could be derailed by sniping from the European Union side, or from British supporters of Brexit, or both. It is also possible that both sides in the negotiations are accentuating the positive, not wanting to be blamed if the talks ultimately fail.

Nevertheless, both sides appear eager to resolve the protracted crisis over Brexit. Mr. Johnson has promised repeatedly that Britain will leave the European Union on Oct. 31, without an agreement if necessary.

That would seem to be impossible, since the British Parliament has passed a law designed to stop a withdrawal without a deal — a step the majority of lawmakers believe would be hugely damaging — so an agreement would provide a way out for Mr. Johnson.