Reid suggested the bill could bypass some of the procedural hurdles that normally slows legislation.
"I think it would be a waste of the Senate's time to have a motion on a vote to proceed and a waste of time afterwards," he said. "So I hope that we can get on this legislation tomorrow. I doubt if there will be problems on my side. If there is, I will work to clear them."
Sen. Patrick Leahy (D-Vt.) suggested earlier this month that that he would try to bring up his Runaway and Homeless Youth and Trafficking Prevention Act, which aims to prevent youths from being trafficked, when senators take up Cornyn's bill.
McConnell, however, said that he hopes that after time for amendments and debate the legislation sees "strong bipartisan support here on the Senate floor."