COLOGNE, Germany — European leaders should modify rules to include Britain and the United States in their defense-cooperation efforts, ending a simmering dispute that could turn toxic over time, according to the director general of the European Union Military Staff.

“We will find a way [on] how to engage the United States and other third-party states,” Lt. Gen. Esa Pulkkinen told Defense News in an interview in Washington last week. But he cautioned that the unresolved issue could become a “permanent” thorn in the side of relations with the United States, in particular.

At issue are the conditions for access to the multibillion-dollar European Defence Fund and its associated collaboration scheme, the Permanent Structured Cooperation, or PESCO. The funds are meant to nurse the nascent defense capabilities of the continent’s member states, with the idea that NATO would be strengthened in the process.

Officials have left the door open for the U.K., which recently left the EU, as well as its defense companies to partake in individual projects, given the country’s importance as a key European provider of military capabilities. But the exact terms have yet to be spelled out, requiring a balancing act between framing member states as primary PESCO beneficiaries while providing a way in for key allies.

Defense officials in Washington previously criticized the EU initiative, complaining that it would needlessly shut out American contractors. European leaders countered that the program is first and foremost meant to streamline the bloc’s disparate military capabilities, stressing that avenues for trans-Atlantic cooperation exist elsewhere.

“EDF and PESCO isn’t everything in the world,” Pulkkinen said in Washington. “We are not going to violate any U.S. defense industrial interests.

“The defense industry is already so globalized, they will find a way [on] how to work together.”

While European governments have circulated draft rules for third-party access to the EU’s defense-cooperation mechanism, a final ruling is not expected until discussions about the bloc’s budget for 2021-2027 are further along, according to issue experts.

Officials at the European Defence Agency, which manages PESCO, are taking something of a strategic pause to determine whether the dozens of projects begun over the past few years are delivering results.

Sophia Besch, a senior research fellow with the Centre for European Reform, said the jury is still out over that assessment. “The big question is whether the European Union can prove that the initiatives improve the operational capabilities,” she said.

Aside from the bureaucratic workings of the PESCO scheme, the German-French alliance — seen as an engine of European defense cooperation — has begun to sputter, according to Besch.

In particular, Berlin and Paris cannot seem to come together on operational terms — whether in the Sahel or the Strait of Hormuz — at a time when Europe’s newfound defense prowess runs the risk of becoming a mostly theoretical exercise, Besch said.

The EU members’ ambitions remain uneven when it comes to defense, a situation that is unlikely to change anytime soon, according to a recent report by the German Marshall Fund of the United States.

“The dispute around the concept of strategic autonomy has not led to any constructive consensus, and it will likely affect debates in the future,” the document stated. “Member states and the EU institutions will continue to promote different concepts that encapsulate their own vision of defense cooperation.”

Aaron Mehta in Washington contributed to this report.