Sweden's Migration Agency also revealed that an immigration reception centre in Malmö had been forced to close temporarily on Tuesday night and Wednesday morning after it became so overcrowded that staff deemed it to be unsafe.
Around 800 refugees turned up at the temporary accommodation in the Jägersro area of the southern Swedish city.
“Very many people came at once, for safety reasons we could not let everyone come inside. It is only designed for one hundred people,” Matilda Niang, a spokesperson for the Migration Agency's press office told the TT newswire.
The agency said that Malmö alone had processed 1533 asylum seekers over the last seven days, with thousands more seeking asylum in other Swedish towns and cities.
It added that it had transported asylum seekers unable to find shelter in southern Sweden to different parts of the Nordic nation where there was space in emergency housing, such as Örebro in central Sweden.
To put the latest figures into context, Sweden's Migration Agency received a total of 11,743 applications for asylum during the whole of July, up from 6,619 in June. Last week alone there were 5200 new arrivals.
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Refugees have been rushing to get to the Nordic country over the past seven days amid scenes of chaos in neighbouring Denmark, which offers fewer benefits to asylum seekers. Germany has closed its borders after regional officials said that they had been overwhelmed by the latest influx of people fleeing violence in the Middle East and Africa, with Hungary also stopping any refugees from entering its territory.
Sweden, which over the past few years has taken in the second largest number of asylum seekers in the EU after Germany, has vowed to keep its borders open for the time being.
“We are not close to such a situation,” said Prime Minister Stefan Löfven at a joint press conference with European Parliament President Martin Schultz on Monday.
However Finland announced on the same day that it would increase checks at its Lapland border with Sweden after growing numbers of asylum seekers crossed into its territory from Sweden's far north in recent weeks.
Sweden is expected to receive at least 74,000 asylum applications by the end of 2015.
The International Organization for Migration has said that more than 430,000 people have crossed the Mediterranean to Europe this year, with 2,748 dying en route or going missing.