The government's directive on ID checks lapses on July 4th after which an automatic two-week grace period kicks in, the paper said.
Any ruling to re-instate the checks would take effect on July 18th at the earliest.
For asylum seekers without identity documents, this will offer their best chance of getting into the country since Sweden started checking IDs on cross-border public transport at the start of this year.
The end of border checks will coincide with the holiday season, and government authorities are scrambling to prepare for what could be another heavy influx of people seeking shelter in the country.
“For the police to be able to do their job our staff also need holidays, which means that our capacity is lower in the summer months,” border police chief Patrik Engström told Aftonbladet.
“That means that if there is a major influx of people again this summer we will have to bring staff in on overtime and take other measure to ensure secure border checks.”
At the high point of the refugee crisis last autumn Sweden was taking in an average of 9,000 asylum seekers every week. In all, Sweden received 163,000 asylum applications in 2015.
However it has seen the number of migrant arrivals fall sharply since it introduced police border controls in November, followed by passenger ID checks at ferry terminals and Copenhagen Airport's train station at the turn of the year.