The bill also includes an amendment by Sen. Bob Menendez (D-N.J.) on Iran’s jamming of satellite communications.
Reid previously tried to pass similar legislation by unanimous consent in late March.
The agreement also includes additional changes strengthening human rights provisions, recognizing the European Union’s recent cut-off of designated Iranian banks from the Society for Worldwide Interbank Financial Telecommunication (SWIFT), which is used to move money around the world, and refining provisions related to victims of terrorism in Iran, the Democratic aide said.
The bill is similar to a measure reported out unanimously by the Senate Banking Committee in February. It has been languishing for several months as lawmakers weighed Kirk's requests and concerns raised by Sen. Rand Paul (R-Ky.) that the bill could be read as an act of war.
The legislation does add language from Paul, noting that this bill is not an authorization of use of military force.
Timing of passage is still in play as the Senate moves to vote on Federal Reserve nominees at noon.
The improved package has the backing of the American Israel Public Affairs Committee (AIPAC), a pro-Israel lobbying group, which sent a letter to lawmakers on Thursday urging support for the measure.
A Democratic aide said the timing of the bill this week was important for the Senate to send a message it would not back down on sanctions ahead of next week’s nuclear talks between Iran and the P5+1 group, the five permanent U.N. Security Council members plus Germany.
Tehran reportedly wants the sanctions lifted as the second round of nuclear talks get under way, as the Obama administration has credited the pressure from sanctions for bringing Iran back to the negotiating table.
House Foreign Affairs Committee Chairman Ileana Ros-Lehtinen (R-Fla.) expressed concern Thursday about the direction the bill is taking.
“I am deeply concerned that the administration’s foolish embrace of yet another round of negotiations will only embolden the regime," she said at a hearing. "The administration has made concession after concession in its negotiations with Iran only to come empty handed.
"The Iranian approach seems to be: 'What's mine is mine, and what's yours is negotiable.'"
Iran has argued that its nuclear program would be used for civilian, mostly energy, purposes whereas U.S. officials have argued that the nation wants to build weapons.
The House passed its version of the measure in December and passage in the Senate would require lawmakers to agree on a final bill.
Daniel Strauss and Jeremy Herb contributed.