President Barack Obama’s stepped-up push for a criminal justice overhaul is in some ways playing catchup to congressional efforts already well underway.

While Obama pushed for a broad overhaul bill to reach his desk this year in a Tuesday speech to the NAACP in Philadelphia, a working group of senators has been toiling away trying to agree on language — with various House pushes in progress.

Senate Majority Whip John Cornyn, R-Texas, said members of the Judiciary Committee working group put together by Chairman Charles E. Grassley, R-Iowa, are “remarkably close” to a deal on legislation.

“I don’t think it’s overly ambitious to say we can get a product to the president this year,” Cornyn said at a bipartisan conversation on criminal justice moderated by Bill Keller, editor-in-chief of the Marshall Project and former executive editor of The New York Times. The event — part of the inaugural “Across the Aisle” event held by the Edward M. Kennedy Institute for the United States Senate and the Coalition for Public Safety — was also attended by Sens. Sheldon Whitehouse, D-R.I.; Mike Lee, R-Utah; and Cory Booker, D-N.J., who are all pushing for an overhaul.

Cornyn and Whitehouse, both former attorneys general and members of the Senate Judiciary Committee, have introduced the Corrections Oversight, Recidivism Reduction, and Eliminating Costs for Taxpayers in Our National System, or CORRECTIONS Act.

“Our criminal justice system affects all of us whether we go to jail or not because eventually the people that go to jail, go to our prisons, get out,” Cornyn said to the crowd — a statement that Obama later echoed.

“The question is whether they’re going to acquire the skills they need, those who are willing to try to change, to become productive members of society. I believe that there’s a large segment of people who can be helped,” Cornyn said.

Lee, who has co-sponsored a bill reducing mandatory minimum sentences with ranking Democrat Patrick J. Leahy of Vermont backed by the White House, said the two bills could be combined.

“We do have now 13 members of the Judiciary Committee who are co-sponsors of the [Lee-Leahy] legislation. And we’re getting closer and closer to a point where I think we’ll be able to combine a Cornyn-Whitehouse proposal, which deals with criminal justice on the back end, with some variation of the proposals involved in the Lee-Leahy bill,” Lee said.

The support of Lee and Sen. Rand Paul, R-Ky., for reducing mandatory minimum sentences for drug offenders, has won praise from the White House.

Obama cited both Cornyn and Paul in his speech Tuesday.

Meanwhile, Whitehouse said Grassley has been productive in helping promote legislation, “despite his very frequently expressed skepticisms.”

Grassley has been a fierce defender of mandatory minimum sentences for drug dealers although he has been open to making some changes.

On Tuesday, Grassley said he hopes to reach a deal.

“For several months a group of senators, including Republicans and Democrats, has been working in good faith on a possible criminal justice bill that encompasses a range of issues, including both front-end solutions such as safety valves and reductions in mandatory minimums in certain situations, and back-end solutions such as prison reform,” he said in a statement. “Legislating takes a lot of hard work. We’ve had hours of meaningful discussions up to this point, and those of us in the room are committed to trying to reach an agreement that can gain wide bipartisan support.”

Whitehouse told reporters Grassley has two significant concerns. “One is that he disagrees with the Lee-Leahy bill as filed, and the other is that he has a very broad committee with disagreements on this question within the parties on either side in addition to between the parties.”

But, “for all of his own substantive concerns as a chairman, he’s put together a process that I’ve been very pleased with,” he added.

Sen. Jeff Flake, R-Ariz., said a bill might ultimately need to bypass Grassley.

“Well obviously we’d like him to come on board on the bill but it’s not a surprise. We’ll just work to see if we can get support otherwise,” he said of Grassley.

Whitehouse thinks the more pressing obstacle to passing a bill is “the extraneous mischief that inserts itself into the legislative process.” He defined this mischief as “some political thing that has nothing to do with the bill” — a criticism he had of Sen. David Vitter, R-La., trying to add a “sanctuary cities” amendment to the education bill.

It’s also unclear if Obama’s ramping up the spotlight on the issue will help or hurt.

“I’m encouraged by his leadership and grateful for it and hopefully it leads to something really positive,” Booker said.

And Lee said he appreciated the president’s efforts and hopes it will boost momentum for the bills.

“The president has been very supportive of the idea of front-end and back-end reform. … If we can have both of these pass, and ideally together, we’ll be much better off as a result.”

Leahy said in a statement reacting to Obama’s speech Tuesday that reducing mandatory minimum sentences is “the single most important thing we can do” and there has been “enough talk.”

“The question now is whether we have the leadership in Congress to act and pass the meaningful reforms that our country urgently needs,” he said.

But Sen. Jeff Sessions, R-Ala., another Judiciary member, said following Obama’s on law enforcement would be a mistake.

“It’s just not an area that he has expertise in. He sees it at the political level, and not at the reality level,” he said. “We’ve got to be talking at the street level where people are dying from overdoses and being murdered in their communities. All of which is increasing and part of that increase is a function of presidential leadership.”

Concerns over this issue aren’t confined to those in Washington. Republican presidential hopeful and New Jersey Gov. Chris Christie will be giving a speech on criminal justice in Camden, N.J., on Thursday.