Boris Johnson and the EU’s Ursula von der Leyen are in “close contact” as the two leaders try to seal a trade deal before talks break up for Christmas.
Fishing access and fair competition rules remain key sticking points.
Housing Secretary Robert Jenrick told the BBC an agreement was possible before the end of 2020 but further movement was required from the EU.
Time is running out to clinch a deal before the 31 December deadline, when the UK leaves the EU’s trading rules.
Intensive talks have been taking place in Brussels for weeks between the UK and EU negotiating teams, headed by Lord Frost and Michel Barnier respectively, but there has yet to be a breakthrough.
According to EU sources, the UK prime minister and the European Commission president Mrs von der Leyen have been in direct contact in an attempt to break the deadlock before talks are expected to pause for Christmas on Thursday.
Mr Jenrick said a deal was close “in totality” but the two sides were still apart on future fishing arrangements and what happens if the UK diverges from EU employment and environmental standards in the future.
“There is still some way further to go on that on the EU’s part,” he told Radio 4’s Today. “If that happens, there is a deal to be done.”
He said the details of what was agreed mattered as they had to be acceptable to MPs and the British public.
“We want to make sure we have arrangement in place that ensure we continue to be an independent sovereign nation. The PM is alive to the fact that this is an agreement that is likely to last for a long time.”
The UK insists that it must have control of its waters from 1 January and retain a much larger share of the catch from them than it does under the current quota system.
The EU wants to phase in a new fishing system over a much longer period and retain significant access to UK waters for boats from France, Spain and other countries with large fleets.
What happens next with Brexit?
Mr Barnier said on Tuesday that a last push was taking place, amid reports the EU had made a final offer on fishing which would see it accept a 25% reduction in its access to the UK’s market.
One EU diplomat told the BBC that while most issues had been agreed or were close to being settled, differences on fishing access and quotas “remain difficult to bridge” and the UK was “not moving enough”.
If there is no trade deal by 31 December, trade would be governed by World Trade Organization rules and both sides could place import taxes on each other’s goods, potentially affecting prices.
The UK has said it will keep talking until 31 December but was prepared for any outcome, while EU diplomats have suggested the bloc would be willing to continue negotiations into 2021 if necessary.
Asked in a separate interview with Sky News whether there was a link between Emmanuel Macron’s action to shut the border with France and the Brexit negotiations, Mr Jenrick said: “I hope not”.