Parliament speaker Andreas Norlén gave Kristersson, who has headed a caretaker government since losing the September 13th election, one week to try to build a majority administration.
In the election, the centre-left opposition bloc won 176 seats, versus 173 for Kristersson's right-wing bloc, which includes the anti-immigration Sweden Democrats.
The four parties in the centre-left bloc have failed in several attempts to form a government.
Kristersson is also expected to struggle to build a majority.
"I can already see that this is going to be hard. Probably very hard. But maybe not completely impossible," he told reporters in parliament.
"The math is unchanged and the distribution of seats in parliament is unchanged," he said.
Earlier, lawmakers rejected Centre Party leader Elisabeth Thand Ringqvist as a candidate for prime minister in a vote.
That outcome had been widely expected after Thand Ringqvist informed the speaker that she had failed in a bid to form a government of left and right-wing parties.
READ ALSO: What happens next in Sweden's government formation process?
Policy differences between the parties have prevented both wings from building a majority.
While the four centre-left parties hold a majority, the Centre Party has refused to cooperate with the socialist Left Party, whose support is needed to build a left-wing majority.
The Left Party has insisted it will only support a left-wing government if it is included.
"There are a lot of red lines," Thand Ringqvist told AFP on Tuesday.
Sweden's parliament speaker can put four candidate prime ministers to a vote by lawmakers, after which new elections must be called.
Observers have suggested that the parties will feel increasing pressure to compromise on their political "red lines" the closer they get to the risk of a new election.
Tuesday's vote on Thand Ringqvist was the second PM vote for lawmakers.
They voted down Social Democratic Party leader Magdalena Andersson on October 15th, which was also widely expected after she failed to form a government.
The Scandinavian country has become accustomed to lengthy negotiations to form governments. It took 37 days after the 2022 election and 134 days in 2018.
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