LONDON — The new European Parliament that was voted into office over the weekend, despite having a different political makeup, is widely expected to reach a final agreement this year on stricter online privacy rules that have long been in the works.

The rules, which have been discussed since 2012, would be stricter than those in the United States. They would create one law across the European Union to protect several aspects of online privacy, which is enshrined as a fundamental right in Europe, including restrictions on what information could be shipped overseas. And they would impose multimillion-dollar fines on any company that misuses Europeans’ data.

Still, a crucial question remains: Which European regulator will have the final say in enforcing the rules?

Under the proposals, companies will be able to operate throughout the region if they fulfill the requirements — and the interpretation — of European rules from only one country’s privacy authority. Companies currently must comply with the regulator in each of the 28 countries in the union in which they operate.