The House bill aims to reduce recidivism, and among other provisions, would expand education for federal prisoners and require them to be placed in facilities closer to home.
But a source familiar to the process says the bill's passage in the House with 70 percent of Democrats voting in favor of the measure puts pressure on Senate Democrats to support the legislation.
"That says there is massive momentum," the source said. "That means that whatever has been said in an ivory tower doesn't matter. That means Democrats are going to be forced to vote in favor of this bill."
And on the Republican side, the fact that the bill has the backing of the president, the source predicted, will force Republicans to get in line.
"The president needs a victory and this would give him a bipartisan one," the source said. "Kushner is on board, Trump is on board, that will steamroll any Republican opposition. I have no doubt that Trump will sign this bill into law."
But Sen. Grassley remains opposed to the current legislation, because he argues broader reforms are possible.
"You have a president that needs a big bipartisan victory and if we just did what the House did, it would be a spit in the ocean compared to the problem we face," said Grassley, who said he believes Senate Majority Leader Mitch McConnell will not go over his head in bringing legislation to the floor without his support.
"I don’t think he’s going to bring up anything unless he knows he can get it done in two or three days," Grassley said.
Kushner has led a more than year-long effort within the Trump administration on the issue, working behind the scenes within the White House to carve out a middle ground for seeking reforms to the criminal justice system that could be accepted by Attorney General Jeff Sessions, who has rejected efforts to seek broader sentencing reforms.
The House bill’s bipartisan co-sponsors Reps. Doug Collins, R-Ga., and Hakeem Jeffries, D-N.Y., celebrated their legislative accomplishment Tuesday, with Jeffries making clear that he sees the bill as just the first step in a series of reforms he believes are still needed to the nation’s justice system.
“The FIRST STEP Act is not the end. It’s not even the beginning of the end. It's simply the end of the beginning on a journey undertaken to eradicate our mass incarceration epidemic in America,” Jeffries said.
Collins said the ball is now in the Senate’s court as to whether they will seize upon the current political moment to improve the lives of the nation’s incarcerated population.
“Today’s vote answers a question in the House and asks one of the Senate: Given the chance to accomplish good in this moment, will you act today or regret your inaction tomorrow?” Collins asked rhetorically in a statement.
ABC News' Ali Rogin and Ben Siegel contributed to this report.