The anti-immigration party, led by Jimmie Åkesson, would be the choice of 20 percent of people (one in five) surveyed for a poll by YouGov for Sweden's Metro newspaper, while 18 percent of respondents told Novus that they would back the nationalists in research for broadcaster TV4.
Both polls also suggested that the centre-right opposition Alliance parties that formed Sweden's last government currently have more support than the Social Democrat-Green coalition that was formed after the general in September 2014, with both polls showing a 1.4 percent lead for the Alliance.
"It is a rejection of the government and Stefan Löfven, they are not delivering what voters hoped for," political analyst Stina Morian told Sweden's Metro newspaper.
More than 1,000 Swedish voters were quizzed for each survey.
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The polls follow the results of the country's largest biannual political poll, carried out by Sweden's number crunching agency Statistics Sweden, which also reported that support for right-wing parties was growing on June 5th.
But the study reported a more modest increase in support for the Sweden Democrats, with 14.5 percent of some 9,000 people questioned listing the group as their preferred party.
Li Bennich Björkman, a political scientist at Uppsala University told The Local earlier this month that there were a number of potential explanations for the recent rise in nationalist support.
"Media attention - even if it is negative - still makes the party more visible. There has been a lot written and talked about the Sweden Democrats."
She added: "The other thing is that it has become less stigmatized to say that you are sympathetic to the Sweden Democrats, due to their rise in the elections and afterwards."
"Thirdly, their main questions about migration and integration have become really core issues in Swedish politics right now," she added.
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Sweden continues to take in more asylum seekers per capita than any other European Union member state. While the government has said it backs new EU quotas designed to distribute refugees more evenly across the 28 member bloc, it remains strongly in favour of maintaining the country's tolerant and open approach to immigration.
On Thursday, Sweden's Prime Minister Stefan Löfven hit the headlines for failing to send any senior politicians to support the Social Democrats' sister party in Denmark, which is holding a general election.
The Social Democrats insisted that this was due to other commitments but the move was seen by many as a protest against the harsh tone on immigration that has characterized the campaign over the border.