Prime Minister Theresa May insists the agreement setting out the U.K.’s future relationship with the European Union won’t be an off-the-shelf affair. Britain will get a “bespoke” arrangement exclusively tailored to its own interests, she has said.

That may be true. It is unlikely a future agreement will look like the EU’s close economic relations with Norway, which has membership in the bloc’s single market and puts no caps on EU migration. Nor will it simply mirror the EU’s new trade agreement with Canada, for example.

Officials in Brussels appear to agree. If the U.K. wants a tailor-made deal, they say, there is no reason why it can’t have one.

But the apparent accord on this point masks a fundamental difference between the two sides. British politicians and officials argue the U.K. brings things to the EU that no other partner does. They cite security and foreign-policy assistance, counterterrorism and intelligence sharing as well as the size and proximity of its economy. They see those assets as justifying a special deal that reflects the U.K.’s singular pattern of EU cooperation, which no other country has.

Yet for now—a time when with opening positions still being formed—that argument isn’t carrying weight in Brussels and other EU capitals.

For the EU, a bespoke agreement means Britain would end up in a unique place on the spectrum of rights and obligations Brussels has long used to map out ties with non-EU countries.

Such an agreement could be narrow or broad, addressing any or all of the following, and more: tariff levels and nontariff barriers, trade in services, capital- and labor-market issues, business regulation, agriculture, fisheries, and noneconomic issues such as database-sharing to help fight terrorism.

But EU officials insist the U.K.’s access to the EU market will depend on the obligations it is willing to undertake.

Until now, Mrs. May has only given hints about her vision of the future relationship. She is expected to set out more details before the end of March, when she triggers Article 50 of the Lisbon Treaty and starts a two-year time period for the two sides to agree on Britain’s exit terms.

She has said she wants the U.K. to control immigration from EU countries and end the European Court of Justice’s jurisdiction over U.K. affairs. She has also said she wants Britain to enjoy free trade in goods and services and for British companies to have maximum freedom to operate in the single market.

European governments say restricting free movement of labor and ignoring the ECJ are incompatible with membership in the EU’s single market. For one thing, the ECJ polices the single market; if the British aren’t subject to its rulings, how can they be part of that market?

British government ministers have openly disagreed about whether the country should keep making payments to the EU, as Norway, Switzerland and even Israel do. EU officials say Britain will have to pay into the budget if it wants access to EU programs on science and research, education and security. In other words, Britain can’t have its cake and eat it, too.

Vicky Ford, a member of the European Parliament from Mrs. May’s own Conservative Party, said in a speech this week that Mrs. May had outlined three priorities: immigration control, market access and control over regulation. She urged both sides to remain flexible and said the U.K. should steer clear of maximalist demands, especially on immigration.

“Like a three-legged stool, if one is prioritized more than the others, it can all become unbalanced,” she said.

Yet, so far, no dice. On Tuesday, the EU’s chief Brexit negotiator Michel Barnier gave the clearest indications yet that Britain will have to base its choices on a largely predetermined set of benefits and costs that already apply in other EU agreements.

“When you examine different options for future relations—naturally, you don’t need me to do it—you can look at the different models that exist,” he said, in his first public remarks since taking up his post. He cited as an example Norway, whose close access to the single market is secured by large payments to the EU.

Mujtaba Rahman, Europe head for the Eurasia Group consulting firm, said Mr. Barnier has made “implied and assumed” thinking “an explicit part of the EU’s red lines.”