Germany lashed out at Turkey for jailing a German national and suggested the latest escalation in a growing feud between the two countries could cost Turkey billions of euros in European assistance.

Wednesday’s threat came hours after Berlin summoned Turkey’s ambassador to protest this month’s arrest of German human-rights activist Peter Steudtner under Ankara’s controversial antiterror laws.

The latest development in a lengthening string of disputes between Germany and Turkey, both members of the North Atlantic Treaty Organization, is putting pressure on German Chancellor Angela Merkel to put a stop to what critics of Turkey see as increasingly brazen provocations.

German Foreign Ministry spokesman Martin Schäfer said the Turkish ambassador was “told very clearly that the arrest of Peter Steudtner and other human-rights activists is neither understandable nor acceptable.”

Ms. Merkel’s spokesman said Turkey’s accusations against the activist were a “transparent attempt to discredit and criminalize dissenters.”

The spat broke out on July 5, when Turkish police arrested 10 Amnesty International activists, including Mr. Steudtner, who had gathered in Turkey for what the organization called a routine workshop.

Six of the rights activists, including Amnesty’s Turkey director, have been jailed pending trial on charges of aiding a terror group. Prosecutors alleged that the director was linked to three unrelated terror organizations. The other four activists previously detained are on bail facing investigation.

“These activists are innocent. The decision to proceed shows that truth and justice have become total strangers in Turkey,” said Amnesty International’s secretary-general, Salil Shetty.

“We firmly believe that this arrest is absolutely unjustified,” Ms. Merkel said late Tuesday. “We declare our solidarity with him and the other detainees and the German government, at all levels, will do everything we can to secure his release.”

Spokesmen for Ms. Merkel and for German Foreign Minister Sigmar Gabriel said the government would consult on possible further action.

Ms. Merkel’s spokesman said the European Union was due to re-examine this year its €4.5 billion ($5.2 billion) in aid to Turkey earmarked for measures to strengthen democracy, the rule of law, economic growth and competitiveness as part of the country’s longstanding bid for EU membership.

“We think it would be right to review this [aid] now given the latest developments in Turkey,” the spokesman said. It is unclear “whether this aid can achieve the desired result given the current situation,” he added.

A spokesman for Turkey’s Ministry of Foreign Affairs didn’t respond to a request for comment.

As the rapport between Berlin and Ankara has deteriorated, Ms. Merkel has sought to avoid an escalation, limiting herself to verbal protestations.

Berlin’s room to maneuver is limited. Germany relies on a pact with Turkey that has drastically reduced the inflows of migrants from the Middle East since the summer of 2015, which caused a political crisis for the chancellor because Turkey is the main transit route to Europe.

Berlin officials have also been reluctant to take action that could further endanger nine other Germans detained since last July in Turkey under antiterror laws. And they are wary of inflaming passions among Germany’s three million ethnic Turks—the largest Turkish diaspora in the world, where support for the Turkish regime is strong, according to recent surveys.

Also, the German military is stationed in Turkey as part of the international alliance against Islamic State in Syria and Iraq.

But the approaching September election in Germany, when Ms. Merkel will be seeking a fourth term, has made it more difficult for her to be seen as refusing to push back against what many in the West perceive as Turkish President Recep Tayyip Erdogan’s increasingly authoritarian drift at home.

“This is a huge dilemma. Voters are clearly in favor of taking tougher actions against Erdogan…and if Erdogan continues to behave this way it can damage her during the election campaign,” said Oskar Niedermayer, professor of political science at Berlin’s Free University.

A recent survey conducted by YouGov showed that 84% supported the German government in banning Mr. Erdogan from speaking to supporters on the sidelines of the Group of 20 leading economies summit in Hamburg this month.

Martin Schulz, Ms. Merkel’s main rival for the chancellery, on Tuesday said German citizens were “at risk of becoming a hostage to the politics of President Erdogan.”

Mr. Steudtner’s arrest is just one of the many Turkish actions that have caused outrage in Germany.

Deniz Yucel, a prominent German-Turkish journalist arrested in Istanbul in February under terrorism suspicions, remains in detention without charges despite repeated protests from Berlin. Mr. Yucel has denied any wrongdoing.

And after Turkey repeatedly barred German lawmakers from visiting troops stationed at the Incirlik air base in the country’s south, Berlin said it would relocate the contingent to Jordan.

Ankara has previously said it had banned all foreign civilian dignitaries from visiting the base in Incirlik for security reasons.

This month, Turkey began barring lawmakers from visiting a separate German contingent at the NATO air base in Konya, prompting parliamentarians to demand its withdrawal, too. The visits, they argue, are part of the mandates governing German military deployments abroad.

A German pullout from Konya would be far more disruptive to the fight against Islamic State than the Incirlik move. Germany provides about a third of the crews in Konya that operate and maintain the Awacs surveillance aircraft seen as crucial in the campaign.

Turkey has accused Germany of being a haven for terrorists wanted by Ankara and has protested its decision to grant asylum to several individuals it believes were involved in last year’s aborted coup attempt against Mr. Erdogan.

A German official on Wednesday said Ankara had provided Germany with a list of prominent German companies it says support terrorism, including car maker Daimler AG and pharmaceutical company BASF SE .

A spokesman for BASF declined to comment. Daimler spokeswoman Ute Wüest von Vellberg said the company hadn’t seen the list and declined to comment further.

A spokeswoman for ​Germany’s BKA Federal Criminal Police Office confirmed that Turkey has provided it with a list consisting of what it referred to as vague information and accusations about companies. But she also said no direct threat to the companies resulted from the list and it has also currently no information about specific threats to these companies’ subsidiaries in Turkey.