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BERLIN—Germans go to the polls in three state elections on Sunday that will allow voters unhappy with Chancellor Angela Merkel’s welcoming refugee policy to vent their frustration.

The elections could see the anti-immigrant Alternative for Germany party make its biggest gains since its founding three years ago, polls show.

The party’s surge could reduce the sway of Ms. Merkel’s conservatives in state legislatures and put renewed pressure on the chancellor to shut Germany’s doors to migrants—a step she has so far rejected despite about one million arrivals last year. 

Campaigning on the home stretch this weekend, Ms. Merkel acknowledged problems in the government’s handling of the migration crisis but urged worried voters to pick her conservative party nevertheless.

“We have lost time in the question of asylum, of the Geneva Convention, in the question of giving protection to those who need protection but telling those who don’t need protection, ‘You can’t come to our country,’” Ms. Merkel said at a rally Saturday in the southwestern state of Baden-Württemberg. “Those who want fast decisions must vote for the Christian Democratic Union.” 

Ms. Merkel’s CDU is facing its toughest battles in the two former West German states that are voting, Rhineland-Palatinate and Baden-Württemberg. In the former, one of the CDU’s rising stars, Julia Klöckner, is trying to reverse a late slide in the polls. In the latter, the CDU has fallen behind the left-of-center Greens and their popular state premier, Winfried Kretschmann. In both, the Alternative for Germany—known by its German initials AfD—is polling around 10%, double the threshold to enter the state parliament.

Ms. Merkel’s party appears assured of victory in the former East German state of Saxony-Anhalt. But recent polls have the AfD getting 18% of the vote—putting the upstart party within striking distance of attaining the second-most seats in the legislature. The party wants Germany to close its borders to asylum applicants.

A strong result for the AfD wouldn’t have an immediate impact on how the German government manages the migrant crisis, since the key decisions are made on the federal level and not by the states. But it would represent the strongest electoral rebuke to Ms. Merkel since she decided in late summer to welcome the hundreds of thousands of refugees and migrants trekking to Germany, rather than to wall her country off. Such a result, analysts have said, could lead to increasing questions inside her party as to whether Ms. Merkel may be a liability at the top of the ticket in national elections next year.

To be sure, Ms. Merkel remains in a strong position even though the migrant crisis has taken a toll on her popularity. Her approval rating stands at 54%, according to pollster Infratest Dimap—a decline from 67% last summer but still high compared with the leaders of many other European democracies. The CDU is the most popular party in the country, polling at 36%, compared with 23% for the left-of-center Social Democrats, the junior partner in Ms. Merkel’s governing coalition.

Ms. Merkel has also responded to German voters’ concerns by promising to substantially reduce the number of asylum-seekers arriving in Germany from last year’s total of roughly a million. While she has refused to close the country’s borders to migrants—warning of devastating consequences for Europe’s system of passport-free travel—she has toughened asylum laws and pushed Turkey to prevent people from getting into boats bound for Greece.

But in an indication of the jitters among Ms. Merkel’s allies, Interior Minister Thomas de Maizière, also of the CDU, said in an interview published Saturday that the AfD “damages our country.”

“The AfD doesn’t have a political plan or the right answers,” Mr. de Maizière told newspaper Die Welt. “Even those who vote for it don’t think the party has better policies or can contribute to the solution. It’s just about teaching ‘them up there’ a lesson. That’s not sustainable for long.”