It would also extend retroactively a reduction in the sentencing disparity between crack and powder cocaine signed into law in 2010, which could affect thousands of drug offenders serving lengthy sentences for crack-cocaine offenses, which were dealt with far more harshly than the same crimes involving powder cocaine. That disparity hit black Americans hard while letting many white drug dealers off with lighter punishments.

The other half of the proposed bill creates a package of incentives and new programs aimed at improving prison conditions and preparing prisoners for re-entry into their communities. It would require the Justice Department to create a risk and need assessment system to nudge prisoners toward better outcomes. And it would expand time credits that reward good behavior and create new ones for participating in job-training and other programs that allow offenders to reduce their time behind bars.

The legislation would also improve conditions for incarcerated women, prohibiting the shackling of female inmates while pregnant, and would require the Bureau of Prisons to locate prisoners in facilities close to their homes, if possible.

The Fraternal Order of Police, the country’s largest police organization, said last Friday that it would support the bill, and the National Sheriffs’ Association appeared to have dropped some previous objections after exceptions were made to block certain fentanyl offenders from eligibility for “good-time credits” included in the prison overhaul portion of the bill.

But powerful pockets of opposition remain among some law enforcement officials and conservative lawmakers — like Senator Tom Cotton, Republican of Arkansas — who argue that sentencing changes like those proposed pose a risk to public safety. However, they lost a powerful ally within the administration when Mr. Trump fired his attorney general, Jeff Sessions, last week. Mr. Sessions’s temporary replacement, Matthew G. Whitaker, has signaled that he is more open to the changes.

Mr. Trump himself is leery of appearing weak on crime, and he has been susceptible to arguments from opponents of a sentencing overhaul that endorsing one could arm his critics. Still, Mr. Kushner has pressed the issue for months, and some of the president’s advisers say they think the effort could help improve his anemic standing with African-American voters, even if only marginally.

In his remarks on Wednesday, Mr. Trump tried to address both points, saying that the legislation would be tougher on hardened criminals. But in a reference to the tough-on-crime policies embraced by President Bill Clinton, Mr. Trump also said that the legislation would begin to roll back portions of the “Clinton crime bill” that had a “very disproportionate and very unfair” effect on black Americans.