Brexit talks risk reaching a stalemate if there is no progress in the next round of negotiations between the European Union and the British government, EU sources have said. The two sides are due to resume talks next week, the final round scheduled before a “high-level conference” in June to assess progress before the end-of-year deadline. After the chief negotiators, Michel Barnier and David Frost, exchanged testy letters last week, a senior EU official said there was a risk of stalemate if the EU did not see progress on its vital interests, including how to ensure fair competition, or a level play field, between British and EU companies under a free-trade deal. “That is a crucial round,” the official said. “If there is no parallel progress – level playing field, protection of fundamental rights and governance – then we risk going into July with a major problem, the stalemate Barnier warned about.” Earlier this month, the EU’s chief negotiator said “parallelism is a condition for progress”, meaning that talks had to advance in all areas, including priorities for the EU, such as the level playing field and an agreement on fishing rights. Frost argues the UK has set out a “comprehensive set of proposals” to prevent unfair competitive advantages, but multiple EU sources say the British ideas are inadequate. In a setback for Frost, EU diplomats have rejected calls for the union to change its approach to the negotiations. One of several diplomats who spoke to the Guardian and were opposed to the idea of rewriting the negotiating mandate, said: “A large majority of member states, maybe all of us, think that we have to stick to the mandate at this point of time and I don’t see a lot of movement … there are not many lights in the tunnel.” In a further blow to the UK, diplomats also backed Barnier’s accusation that the British were trying to “cherrypick” benefits from the EU single market that were not available to other countries with a free-trade agreement, such as Canada and Japan. “The demands of the UK government at the moment are very unilateral,” said a second EU diplomat. “They want to keep the things that are suiting them, but give away things in our interest.” The diplomat referred to British interest in an agreement on professional services, such as law and auditing, that goes beyond what the EU has done with other trade partners. “To have British lawyers providing services around the EU, I am not sure that is of interest to our lawyers, especially when you [the UK] don’t want to be in the common ecosystem [of] common controls and rules.” EU diplomats are tired of the repeated reminders from Frost’s team about the importance of UK sovereignty, pointing out that their governments are also sovereign. “The EU accepts that the UK is sovereign, but that is not the issue,” said the EU official. “The issue is what you as a country are going to commit to to get access to our market. That is the real question.” A UK government spokesperson said it was to be “expected at this stage in a very difficult negotiation that both sides are making their case robustly and ensuring it is understood. We will continue to approach negotiations constructively, but our position hasn’t changed – we won’t agree to any EU demands for us to give up our rights as an independent state. “And we’ve never asked for anything special, bespoke or unique – we’re looking for a free-trade agreement, based on precedent and similar to those the EU has already got with other countries like Canada.” Adding to pressure on Barnier not to concede on fishing rights, on Monday night the European parliament’s fisheries committee threatened to veto any deal with the UK that did not include a “balanced” agreement on fish quotas, allowing EU fleets continue access to British waters. “No fisheries agreement means no post-Brexit agreement,” said François-Xavier Bellamy, the French centre-right MEP – and member of Barnier’s Les Républicains party – who drew up a report that was adopted with near unanimity by the committee.