ISTANBUL — Prime Minister Recep Tayyip Erdogan offered on Wednesday to hold a referendum to decide the fate of the park in central Istanbul that has become the locus of protests against his government and presented him with the most serious challenge he has faced in his decade in power.

But even as Mr. Erdogan seemed to indicate an attempt at compromise with the referendum offer, he coupled it with tougher language, saying that he ordered his interior minister on Wednesday to end all antigovernment protests within 24 hours.

The referendum was proposed after Mr. Erdogan met with a group of protesters in Ankara, the capital. It was the latest move in a seemingly confused strategy by the prime minister to resolve the crisis over Gezi Park in Taksim Square, with measures that included blaming a roster of supposed scapegoats, including the international news media, foreign financial interests and terrorists.

As Mr. Erdogan’s beleaguered government made its latest offer, thousands of protesters returned to the square after the riot police dispersed crowds Tuesday night and Wednesday morning with tear gas and water cannons.

In Gezi Park, which protesters are campaigning to save in the face of government plans to raze it and build a mall designed like an Ottoman-era army barracks, protesters on Wednesday night were girding for a police raid that they expected to come either overnight or Thursday morning. Many were skeptical of the government’s plan to hold a referendum, and some said it was not a suitable solution.

“They first tell us to go home, and then they present the idea of the referendum?” asked Bora Ekrem, 24, a student. “How can we trust them? If they were sincere about a vote, they would not ask us to leave the park. We will not leave until they declare the park is ours.”