CHICAGO — This city moved Monday toward paying more than $6 million in settlements to the families of two men who died after being taken into police custody, frustrating City Council members who said such cases showed the need for urgent changes to police training and discipline.

“We come here settlement after settlement after settlement, and we never hear termination,” said Alderman Anthony A. Beale, who represents part of the South Side.

“We’re spinning our wheels here. We need to start firing some of these people.”

The estate of Philip Coleman, who was recorded being subdued with a Taser by Chicago police officers and dragged from a cell in handcuffs hours before dying at a hospital in 2012, would receive $4.95 million if the agreement is approved by the full City Council on Wednesday. The estate of Justin Cook, who died of an asthma attack after a foot pursuit led to his arrest in 2014, would receive $1.5 million. Several witnesses said the police refused to let Mr. Cook use his inhaler despite repeated pleas that he could not breathe, a city lawyer told aldermen on Monday.

In both cases, the city’s Independent Police Review Authority has an open investigation, though no officers have been fired or criminally charged.