J.P. Morgan Chase & Co. has offered to pay about $3 billion as it seeks to settle criminal and civil investigations by federal and state prosecutors into its mortgage-backed-securities activities, according to a person familiar with the discussions.

The Justice Department rejected that sum as billions of dollars too low for the number of cases involved, according to the person. But the discussions have widened to include other investigations of J.P. Morgan, and the final tally could be larger, the person said.

A settlement on residential mortgage-backed securities would solve one of the bank's biggest legal problems. The largest U.S. bank by assets faces at least seven Justice Department probes, on matters ranging from energy trading to its hiring practices in China.

The offer from the bank shows that its top executives and board are weighing the time and effort needed to fight, as well as the impact on the bank's reputation and employee morale, according to people familiar with the bank's strategy. At the same time, they are leery of agreeing to settlement terms that could open them up to additional private litigation, the people said.

A spokesman for J.P. Morgan declined to comment.