The European Union is willing to accept UK demands that post-Brexit fishing opportunities be divided using a scientific method that reflects the number of fish in UK waters, rather than the EU's Common Fisheries Policy.
Michel Barnier, the EU's chief negotiator, said Brussels could agree to the use of zonal attachment, a key British request in ongoing trade negotiations, if it was coupled with other factors such as assessing the economic impact on coastal communities.
Zonal attachment will benefit UK fisherman because more fish have moved to British waters as a result of climate change since Common Fisheries Policy catch shares were set in the 1970s and 1980s.
The EU has previously insisted EU boats should have the same access to UK waters as now. and under the same conditions.
Mr Barnier admitted that the EU's position on fishing was "clearly not" balanced when he gave evidence to the House of Lords Select Committee about the post-Brexit trade negotiations.
He said Brussels was willing to be "creative" to break the deadlock, according to a transcript of the June 23 meeting, which was published on Monday.
He said: "I am waiting with much patience for a reply from the British side. If there is no response, there will be no agreement on fisheries and no agreement on trade."
While Mr Barnier ruled out the Norway-style annual negotiations on fishing opportunities Britain wants, he suggested some yearly talks could be possible.
"You can discuss fishing stocks regularly every year in the light of the scientific advice, so that we can protect resources and biodiversity, but negotiating access to waters and the fish in those waters every year would be impossible for 100-odd species," he said.
"There will be no trade agreement with the UK if there is no balanced agreement on fisheries. Is this 'balanced agreement' the British position, as it is now? Certainly not. Is it the European position as it is today? Clearly not."
Mr Barnier said he offered UK negotiators a compromise that "would take account, of course, of the zonal attachment that the UK wants".
"We must take account of that, but of other parameters as well: historic fishing rights, sometimes dating back many centuries; the economic interests of coastal fishing communities in the EU and the UK; and international rules from the UN on biodiversity," he said.
Mr Barnier was speaking before the last round of intensified negotiations in Brussels broke up a day early with what he described as "serious divergences" between the two sides. Talks resumed in London on Monday.