Talks between chief negotiators Lord Frost and Michel Barnier are expected to last late into the night in Brussels as officials stressed there had been no breakthrough in the latest discussions that started just before midday on Saturday.
A Government source said: “Talks are continuing overnight, but as things stand the offer on the table from the EU remains unacceptable.
“The Prime Minister will leave no stone unturned in this process, but he is absolutely clear: any agreement must be fair and respect the fundamental position that the UK will be a sovereign nation in three weeks’ time.”
With the UK teetering on the brink of a no-deal exit, the Government has stepped up preparations for crashing out of the single market when transition arrangements end on December 31.
Reports have also suggested that ministers are considering beefing-up Navy powers in legislation to authorise them to board and arrest fishermen found to be contravening post-Brexit rules.
Tobias Ellwood, Conservative chairman of the Commons Defence Committee, called the threat “irresponsible” while former European commissioner Lord Patten said the Prime Minister’s no-deal rhetoric was based on the “runaway train of English exceptionalism”.
But Admiral Lord West, a former chief of naval staff, said it was “absolutely appropriate” for the Royal Navy to protect UK waters from foreign fishing vessels if asked to do so in a no-deal Brexit scenario.
“The Royal Navy should protect our waters if the position is that we are a sovereign state and our Government has said we don’t want other nations there,” Lord West told BBC Radio 4’s Today programme.
When asked about the UK’s decision to ready Royal Navy patrol ships, an Elysee Palace official in Paris reportedly replied using the British wartime slogan, telling journalists: “Keep calm and carry on.”
The trade talks continue to be deadlocked over the thorny issues of fishing rights and the so-called level playing field “ratchet” that would tie the UK to future EU standards.
Mr Johnson, in a speech at a climate change summit on Saturday, appeared to take a dig at French president Emmanuel Macron over the fishing row.
Mr Macron is said to have threatened to veto a UK-EU deal after expressing dissatisfaction at the new quota terms being thrashed out for French fishermen.
In his closing remarks, the Prime Minister thanked summit co-host Mr Macron, adding that he knew the En Marche! leader “shares my keen interest in protecting the ecosystems of our seas”.
The latest impasse comes as farmers warned there will be “significant disruption” to the sector if the UK fails to reach a Brexit trade deal with the EU.
The National Farmers’ Union (NFU) said it was “critically important” that a free trade agreement was reached between both sides, with a priority on securing a tariff and quota-free deal.
More than 60% of the UK’s agricultural food and drink production – worth £14.5 billion to economy – is exported to the EU, making it the largest trading partner for British farmers.
But without a deal at the end of the Brexit transition period on December 31, farmers could lose free access to the EU “overnight”, the NFU warned.
Reporting by PA