The Social Democratic government and opposition parties said on Friday a deal had been reached and that a press conference would be held later in the day, but gave no more details.
The deal includes the introduction of three-year temporary residence permits, with exceptions for some groups including families with children, unaccompanied minors, according to a source close to the negotiations.
Except for very rare cases, everyone granted asylum in Sweden is given permanent residency, a Migration Agency spokeswoman said, Reuters reported.
On Thursday, the Migration Agency said it needed an extra 70 billion Swedish crowns ($8.3 billion) over the coming two years and that it expected between 140,000 and 190,000 asylum seekers this year.
Finance Minister Magdalena Andersson said across-the-board spending cuts and raised borrowing was needed to cope with the situation but that it would anyway take longer time to get back to balanced public finances.