In the wake of this month’s terrorist attacks in Paris, European leaders are calling for significant changes to what has long been a paradox of their borderless continent: Their citizens can move freely, but information about them does not.

There is no European no-fly list, because there is no European data­base of air travelers. People inside a 26-nation zone can speed from the tip of Portugal to the border with Russia without once having their passports scrutinized. Many E.U. citizens enter and exit Europe without ever being checked against police databases.

The gaps can lead to delayed security responses at best and flawed ones at worst, critics say, and attackers have sometimes exploited the issues to their advantage. Now, after the bloody assaults that claimed 17 victims in Paris and after dozens of suspected Islamist militants were rounded up around Europe, European leaders are pushing to fix what they say are flaws in the system.

E.U. nations plan “to share information, intelligence, not only with the European Union but also with other countries around us,” E.U. foreign policy chief Federica Mogherini said Monday after a meeting on counterterrorism with E.U. foreign ministers and top diplomats from several Middle Eastern nations.

Even as officials have stripped away barriers to free travel among European countries, individual nations have continued to hold sway over their intelligence and law enforcement agencies, and there are relatively few E.U.-wide databases in which information about people is stored. In Belgium, for example, police rely on the honor code when they ask new residents with E.U. citizenship whether they have criminal records in other countries.

A Belgian policeman stands guard inside the national airport on Jan. 17. E.U. leaders have proposed creating an air travel database to help seal security gaps like those exploited by the Paris attackers. (Eric Vidal/Reuters)

The Paris attackers were able to use these gaps to their advantage, counterterrorism officials say. One of them, Amedy Coulibaly, drove his common-law wife and others to the Madrid airport before he embarked on his deadly attacks, allowing them to flee to Turkey without immediately drawing the attention of French authorities, who had been monitoring them domestically.

At least one of the brothers who attacked the offices of a satirical Paris newsweekly had traveled to Yemen for training, authorities say. But without a European database that might have helped track their movements, French officials weren’t easily able to watch their air travel, database advocates say.

Gilles de Kerchove, the E.U.’s counterterrorism coordinator, said he is seeking to require that all passports be checked by computer, which would enable agents to run them against databases and track who enters and exits Europe.

An air traveler database, also known as a passenger name record database, or PNR, “would have allowed the police to detect that one of the two brothers went to Oman and then to Yemen,” de Kerchove said of the men who carried out the attack at the Charlie Hebdo newsweekly. “It would have made it possible to detect [Hayat] Boumeddiene, the girlfriend of Coulibaly.”

The key proposal from E.U. leaders is a European air traveler database that proponents say would allow security officials to track information about fliers across the continent. Some E.U. leaders have sought such a database for years. But the European Parliament, which is charged with adopting E.U.-wide legislation, rejected it because of privacy concerns.

E.U. justice and interior ministers will meet this month to take up new proposals. Air travel databases such as those used in the United States track information including credit card numbers, addresses and passport information.

“We are convinced of the irreplaceable usefulness of this tool, on a European scale, to follow those who go to the theater of terrorist operations to fight, and also those who return,” French Interior Minister Bernard Cazeneuve said last week, after a meeting with several other E.U. law enforcement chiefs and U.S. Attorney General Eric H. Holder Jr.

Other proposals include more-rigorous checks at the E.U.’s external borders, improved efforts to restrict terrorist financing, and more robust inter-European criminal and intelligence databases.

The debate echoes one that took place in the United States after the attacks of Sept. 11, 2001, in which the FBI and the CIA were criticized for withholding information from each other.

But that was in a single country, with a single legal system. The European Union’s 28 nations each have their own laws, police and intelligence agencies. They do not always line up with each other, leaving security loopholes.

“If you sometimes see a suspect is already on the books in one country and somehow he or she manages to get a very nice living in another country, it shows there is a problem,” said Latvian Foreign Minister Edgars Rinkevics, who has a leading role in E.U. foreign policy during Latvia’s six-month E.U. presidency, which began Jan. 1.

Currently, a person taking a plane from Germany to France has to show only a ticket, not any identification, before boarding, for example. But ID would be checked on the return flight from France to Germany, a result of differences in the two countries’ laws.

Some countries are stricter than others about how they check the passports of E.U. citizens when they enter and exit the Schengen Area, the borderless zone comprising 26 nations but excluding Britain and Ireland. More than half the time, border guards merely look at E.U. citizens’ passports to judge whether they are authentic, rather than running them through a computer, said de Kerchove, the counterterrorism coordinator.

E.U. intelligence officials say that despite national borders, they do share significant amounts of information. Even without an air traveler database, Spain quickly determined that Coulibaly’s companion had traveled to Turkey.

But other borders stand in the way of quick investigations.

“When I wanted to make a search in the United Kingdom, you need to comply with its system,” said Jean-Louis Bruguière, a former investigating magistrate who once handled many of France’s top anti-terrorism cases. “The German system is a legal system that’s absolutely different from the French.”

The combination of the European Union’s borderless system, the absence of a passenger database and unpredictability about what information is shared among countries sometimes combines to hamper Europe’s response, said François Heisbourg, a former French diplomat and defense official now at the Foundation for Strategic Research in Paris.

“People are not picked up as quickly as they should be, and sometimes not picked up at all,” he said.

But some critics of the efforts to tighten the security nets say that the focus on counterterrorism misses the root causes of the problem.

“There are many different reasons why radicalization is taking place in our member states, and for tackling this, more data collection is not going to help,” said Birgit Sippel, a Socialist member of the European Parliament from Germany.