The United Kingdom left the European Union in January but the two sides are trying to clinch a deal that would govern nearly a trillion dollars in annual trade before a transition period of informal membership ends on 31 December.
After a brief hiatus when London walked away from the negotiating table, both sides are now meeting daily to try to find common ground.
At stake is the smooth flow of cross-border trade as well as the harder-to-quantify damage that a chaotic exit would do to areas such as security information sharing and research and development cooperation.
Barnier and his EU team will be in London until Wednesday, after which talks will switch to Brussels and continue through the weekend, an EU spokesperson said.
EU diplomats were not expected to be briefed on progress in the latest batch of talks until later in the week.
Johnson told reporters he was very glad to be talking with the EU again, but offered no new clues on the likelihood of a deal: “We’ll see where we go.”
Since talks restarted last week, British ministers have said real progress has been made and that there is a good chance of a deal. On Sunday, Ireland’s deputy prime minister, Leo Varadkar, said a deal to avoid tariffs and quotas was likely.
After some progress on competition guarantees including state aid rules, the hardest issue remains fishing - Johnson has insisted on taking back control over Britain’s waters while the EU wants access.
Although Britain insists it can prosper without a deal, British companies are facing a wall of bureaucracy that threatens chaos at the border if they want to sell into the world’s biggest trading bloc when life after Brexit begins on 1 January.