The offer is modest: 2.1 billion euros for a bank with 34 branches and a balance sheet smaller than many German regional lenders. But Donau Bankengruppe’s bid for Banka Primorska, Slovenia’s third-largest bank, has become the kind of deal that Brussels and Frankfurt watch closely, because it asks a question the euro zone has avoided for a decade: who decides whether a bank in one member state may be bought by a bank in another?

Donau, which is based in Vienna and operates in six Central European countries, made its offer public on Sept. 30 after months of private talks. Banka Primorska’s board has recommended it. The European Central Bank, which supervises both banks, is expected to approve it, and two people familiar with the review said supervisors regarded the combination as straightforwardly positive for financial stability. The obstacle is a 25 percent stake held by the Slovenian state since a 2013 bailout, with special rights that allow the government to veto a change of control.

A senior official in the Slovenian Finance Ministry, who was not authorized to speak publicly, said the government had “no objection in principle to foreign ownership” but wanted guarantees on branches, jobs and lending to small businesses, as well as a commitment that the bank’s headquarters and banking license would remain in Slovenia. Donau has offered a five-year moratorium on branch closures and said it would keep a Slovenian board. It has not agreed to keep a separate license, which would defeat much of the purpose of the deal.

“Every government in Europe says it wants cross-border banks until the bank in question is theirs,” said Maja Kovačič, a banking analyst in Ljubljana. “Slovenia is not unusual. It is just small enough that you can see the whole argument in one room.”

The euro zone’s banking union, created after the sovereign debt crisis, gave the E.C.B. authority over the region’s largest banks and created a common fund to wind down failures. It never completed its third pillar, a shared deposit insurance scheme, and without it national governments still treat their banks’ deposits as a national liability and their banks as national assets. Only four cross-border bank mergers worth more than 1 billion euros have been completed in the euro zone since 2014. Supervisors in Frankfurt have said repeatedly that they would welcome more, and have been ignored just as repeatedly.

Gerhard Pichlmayr, Donau’s chief executive, said in an interview that he had been surprised by the intensity of the response. “We are an Austrian bank buying a Slovenian bank 90 minutes’ drive from our headquarters,” he said. “Our clients already cross this border every day. It is odd that their banks cannot.”

Bank employees are not uniformly reassured. Boštjan Kralj, who represents Banka Primorska’s 1,900 staff for a banking union, said five-year promises had a way of expiring. “We have seen what consolidation means,” he said. “It means the back office moves to Vienna, and the back office is where the jobs are.”

The government is expected to decide before the end of November. Whatever it chooses, the official in the Finance Ministry said, it would be read as a precedent. “That,” the official said, “is the problem with being first.”