WASHINGTON — For several years, a handful of lawmakers in Congress have tried to scale back tough sentencing laws that have bloated federal prisons and the cost of running them. But broad-based political will to change those laws remained elusive.

Now, with a push from President Obama and, perhaps even more significant, a nod from Speaker John A. Boehner, Congress seems poised to revise four decades of federal policy that greatly expanded the number of Americans who are incarcerated, to roughly 750 per 100,000, by far the highest rate of any Western nation.

Senator Charles E. Grassley, Republican of Iowa and chairman of the Senate Judiciary Committee, who has long resisted changes to federal sentencing laws, said he expected to have a bipartisan bill ready before the August recess.

“It will be a bill that can have broad conservative support,” said Mr. Grassley, who as recently as this year praised the virtues of mandatory minimums on the Senate floor.