The two bills advanced from committee last week, but not in time to make it to the floor for a vote.
One would ban physical contact by police that could cause suffocation, an addition to the existing ban on chokeholds that sponsor Kenyatta Johnson said was a direct response to the death of George Floyd in Minneapolis.
"My bill prohibits tactics such as putting a knee on an individual's neck," said Johnson.
The other bill would require public hearings before any changes to the police contract.
Its sponsor, Katherine Gilmore Richardson, said the contract has long been viewed as an obstacle to weeding out bad officers.
"With the voice of the people as part of the conversation, we'll be able to hold our police department accountable," she said.
Police commissioner Danielle Outlaw agreed, testifying that arbitrating disciplinary decisions hinders change. She offered a wish list of what she'd like to see in a new contract.
"My top priorities would be including management rights, making sure we have the ability to strengthen our accountability mechanism and then lastly, the arbitration piece is key," she detailed.
The Fraternal Order of Police did not testify. Through a spokesperson, Gilmore said she met with the union separately and they showed no interest in testifying. An FOP spokesman says they weren't invited.
Before recessing, Council passed a residency requirement for new officers and two charter change referenda that will be on November's ballot, ending illegal stop-and-frisk and creating a police oversight commission.