BERLIN—The German state on the front line of the country’s migrant crisis has set Angela Merkel a deadline of Sunday to put an end to the influx or risk unspecified emergency measures, underscoring growing impatience among the German chancellor’s political allies as the flow continues unabated.

On Tuesday, Ms. Merkel dismissed Bavaria’s threat and pointed instead to long-term efforts to negotiate an international solution, setting the stage for a possible confrontation between the government and her increasingly irate allies this weekend.

Figures provided by Bavaria’s interior ministry indicate that the number of migrants who entered Bavaria in October should surpass the 170,000 recorded last month after Ms. Merkel decided to let migrants stranded in Hungary travel to Germany, effectively suspending European Union rules mandating that refugees apply for asylum in their first port of entry in the bloc.

“We have reached a point that I have feared for weeks,” Bavarian state premier Horst Seehofer was quoted as saying in Tuesday’s Passauer Neue Presse daily. “Unfortunately, it’s much worse, it’s so bad that even Bavaria couldn’t have predicted it.”

He set Ms. Merkel a deadline of Sunday. Although states have no power over migration policies, legal experts say Mr. Seehofer could invoke the constitution to put emergency measures in place, arguing that the federal government has failed in its duties to protect the country’s borders.