The king of Denmark has asked a centre-right politician to try to form a new government after the prime minister, Mette Frederiksen, has failed to put together a ruling coalition.

The announcement on Friday night shook the political establishment as Frederiksen has been a staple of Danish politics for decades. Her left-leaning party, the Social Democrats, won the plurality of votes in parliamentary elections in March.

But despite winning the most votes, it was the Social Democrats’ worst electoral showing since 1903 and no party won a majority.

Frederiksen has since tried to form a left-leaning government with the support of Lars Løkke Rasmussen’s centre-right moderates. He is seen as a kingmaker owing to his position in the middle between Frederiksen and Troels Lund Poulsen, the chair of the centre-right liberal party.

However, Frederiksen struggled to gather the support of Denmark’s increasingly fragmented parties and on Friday night, Rasmussen, the country’s former foreign minister, walked out of negotiations and threw his weight behind Poulsen.

King Frederik then asked Poulsen to try to build a new government. The king issued a statement requesting that Poulsen “lead the negotiations with a view to forming a government”, without the participation of the Social Democrats and moderates.

Poulsen will have to marshal a coalition of rightwing parties to form a government – a fragile process that could take weeks. Already, the attempt to build a government has been the longest in Danish history.

Danish politics has skewed increasingly right in recent years, with the March elections showing gains for several rightwing parties. The traditional far-right party, the Danish People’s party, had a particularly strong showing, tripling its votes from the last election to 9.1%.

Immigration has become a hot topic in Denmark, as in other European countries, and even left-leaning parties have adopted stricter policies on immigration control.

The king’s statement indicated that the Danish People’s party had pushed for Poulsen under the condition that the new government had “the explicit goal of introducing measures that will lead to Muslim net-exodus of Denmark”.

After a meeting with the king on Friday afternoon, Frederiksen indicated that there was a growing possibility of a rightwing coalition governing Denmark.

Frederiksen said: “The Danes … have composed the [parliament] in such a way that a rightwing government can absolutely be formed. It might very well be that what we are seeing now is in fact the beginning of that.”

Frederiksen is popular for her handling of Donald Trump’s attempt to acquire Greenland, resisting intense pressure and threats from the US administration. However, Frederiksen performed much poorer on domestic issues, such as taxation and immigration.

Analysts said that while Frederiksen was down, she was not out. If Poulsen failed to pull together a coherent coalition among the rightwing parties, the prime minister could return with a coalition of her own.