At a press conference on Monday, Kristersson maintained that he will want to govern Sweden on an Alliance ticket, that means with policies that are based on the four centre-right parties that used to be in government 2006-2014: the Moderate Party, the Christian Democrats, the Centre Party and the Liberal Party.
"I want as much of the policies of the Alliance as possible, with as many Alliance parties as possible," he said.
"If it is not possible to form this government with all four parties, I want three parties, or two parties in the government, and a good cooperation with that or those Alliance parties that in this case would be outside the government," he said.
The trouble for Kristersson is that the four Alliance parties do not have sufficient support in parliament by themselves, and the Centre Party and the Liberal Party have said they do not want to enter into a government that is dependent on the tacit support of the anti-immigration Sweden Democrats.
On Monday, Kristersson tried to bridge this crack in the Alliance by offering a benchmark evaluation in a year, where the four parties together would assess how the new government has managed to represent the policies of the Alliance as a whole. He also proposed a parliamentary review of new immigration laws, which should result in policies that have broad support in parliament, and by implication would limit the influence of the Sweden Democrats.
But the initial reaction from the Liberal party and the Centre Party on Monday was cool. Jan Björklund, leader of the Liberal party, said on twitter that it was "surprising" that the Moderate Party had rejected the option of letting the leader of the Centre Party explore the possibility a co-operation between the Alliance and the Green Party.
"The Alliance needs to co-operate with another party in order to be able to govern. It is strange to reject the Green party and instead take the risk of being completely dependent on the Sweden Democrats," Björklund wrote.
And in an interview with the news agency TT, the Centre Party's Annie Lööf demanded an answer from Kristersson how one can govern with the active support of the Sweden Democrats. "He must respond to the fact that each budget will need that the Sweden Democrats press the green yes-button," she said.
Lööf did however not reveal how her party will vote - or not vote - in parliament next week. In the past, Lööf as well as Björklund of the Liberal Party have said they would vote against a Kristersson-led government that is dependent on the support of the Sweden Democrats. To TT she now said that their objections remain, but that they will consider the matter again ahead of the vote.
Meanwhile, the leader of the Sweden Democrats, Jimmie Åkesson, said his party would want "guarantees" from Ulf Kristersson, if they are going to vote for him. "The easiest way to do this would of course be talks, but if Kristersson knows another way to do this, I am open to that," Åkesson told the tabloid Aftonbladet.
To be elected as new prime minister, a person does not need to have the support of a majority in parliament, it is sufficient that not a majority does not vote against him or her as prime minister. For a budget to be passed, it requires that there is not an alternative budget that will get more support in parliament.