If the court rejects the case, the FBI has to throw out anything it's collected in that emergency period. The court has rejected surveillance requests in 11 out of more than 33,000 times.
Under Feinstein's proposal, FBI agents would not have to scramble to request permission to turn on surveillance. It would continue while the FBI built a case to present to the court.
"If the court order is not issued, all collection after the time the target is known to have entered the U.S. must be deleted," she said.
The House lawmakers want to craft a similar exception that could allow the FBI in rare cases to seamlessly pick up surveillance, especially in cases of "imminent danger" in which the NSA and FBI believe the suspect is part of a plot to kill Americans.
The FBI has traditional law enforcement techniques it can use to track the suspect, but "electronic surveillance is the most effective way to track both their movements and their communications," said another supporter of changing the law, Rep. Jim Langevin, D-R.I., a member of the House Intelligence Committee.
As they craft the exception to bridge the gap, Langevin said they would also be working to include "a process with checks and balances in there so that this authority could not be abused."
The NSA, FBI and Justice Department would not comment on the proposals.
Associated Press writer Donna Cassata contributed to this report.
Follow Dozier on Twitter at http://twitter.com/kimberlydozier