The UK is still optimistic about striking a trade deal with the EU but has warned that talks could collapse in June unless Brussels abandons its demands for a common fisheries policy and a level playing field, a source close to the UK’s negotiating team has said.

The source said only “limited progress in bridging the gaps between us” had been made at last week’s talks, but there was “confidence that progress can be made quite quickly”.

The two sides will have to come closer to an agreement before a high-level meeting between ministers planned for June if the UK is to stay at the negotiating table.

Boris Johnson is expected to take a more active role in trying to help unblock talks if that date approaches with no breakthrough.

The source close to the UK’s negotiating team said last week’s talks, conducted by video conference, worked well but it was more difficult to build up a personal rapport when communicating remotely.

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He said he was “quite positive” about the probability of a trade deal before the end of the year, when the UK’s transition period ends, echoing the prediction of Michael Gove, the Cabinet Office minister, that the chances of a deal were “definitely better than two to one”.

Responding to those claims about the chances of a Brexit deal, an EU diplomat said the UK was taking a high-stakes gamble.

“Everybody knows that the UK is the world of betting men, but betting alone won’t result in a trade agreement. The continent is as engulfed by Covid as the UK, so I wouldn’t bank on leaders having much time to cater to British fancy,” the diplomat said. “Recovery comes first and last, and a trade deal with the UK is of course a necessary element, but not one that needs to be addressed urgently.”

A second EU diplomat said: “If No 10 doesn’t change its negotiating approach, we will very likely be looking at a no-deal scenario. So we take this statement [on the positive chances of a deal by Michael Gove] as a welcome sign that the UK will change its negotiating stance and start talking on the basis of the joint political declaration,” the document agreed between the EU and Boris Johnson last October.

However, the UK source said: “What is slowing us up is the EU’s insistence on extra provision, notably the level playing field area, aspects of governance and, of course, there is no meeting of minds on fisheries. If they continue to insist on their position on a so-called level playing field, and on continuing the common fisheries policy, for example, we are never going to accept that.

“Draw your own conclusion from that, but I hope they will move on. There are some fundamentals that we are not going to change, we are not going to move on, because not so much that they are negotiation positions as they are what an independent state does.”

He added: “I am confident we will get over that … but probably a bit more noise has to happen before we get to that point.”

Johnson has been facing calls to consider asking for an extension to the transition period to avoid the UK having to trade on World Trade Organization terms with the EU from the end of the year. However, No 10 has said an extension is not up for negotiation.

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The UK source also said it was “still entirely possible to conclude negotiations on the timetable that has been outlined”.

EU officials do not share London’s optimism. Last week the bloc’s chief negotiator, Michel Barnier, accused the UK government of running down the clock by “slow[ing] down discussions on important areas”.

An EU source said: “We are really not making any progress,” describing the latest series of talks as “a standstill round”. While this limited progress would be unsurprising in “normal” trade talks, the accelerated timetable and political attention being consumed by coronavirus has raised questions in Brussels about striking a deal by the end of the year.

“If we had all the time in the world, I really wouldn’t be worried about it,” said the source. “But we are already [almost] into May now. Ratification is going to be extremely complex at the end of the year, so we need more time. We can’t plan for the speedy ratification we were hoping for, for many reasons, and everyone is distracted.”

Brussels officials also perceive a lack of top-level political engagement in the negotiations.

The EU has accused Johnson of backsliding on promises to uphold common standards on the environment, health and workers’ rights – the so-called level playing field. Under pressure from coastal member states such as France, Ireland, Belgium and the Netherlands, Barnier has also insisted there will be no trade deal without an agreement on fishing rights.

Meanwhile in Northern Ireland, the EU-UK specialised committee for the region met for the first time, on Thursday. The DUP MP Ian Paisley accused the group of being “stuffed with civil servants” and bureaucrats. He said: “They are not getting the nitty gritty of business needs.”

The committee will be critical to the future of Northern Ireland as it has the mandate to work out and recommend what checks should be in place in the Irish Sea when the transition period ends in January.

Industry leaders in the region warned that Brexit checks would be imposed on businesses struggling for survival in the Covid crisis, and there was a need for the government to be transparent and straight with them.

Aodhán Connolly, the director of the Northern Ireland Retail Consortium, said there could be “no hail Mary pass, no gentleman’s agreement” on Brexit checks and the detail needed to be “nailed down” soon to give businesses any chance to prepare.