The Senate tried to pass the Cybersecurity Act of 2012 this summer, but several Republicans pulled support for the bill after the U.S. Chamber of Commerce complained that it put too many requirements on the private sector. Since then, Sens. Joe Lieberman (I-Conn.) and Susan CollinsSusan Margaret CollinsDemocrats search for 51st net neutrality vote Overnight Tech: States sue FCC over net neutrality repeal | Senate Dems reach 50 votes on measure to override repeal | Dems press Apple on phone slowdowns, kids' health | New Android malware found Overnight Regulation: Dems claim 50 votes in Senate to block net neutrality repeal | Consumer bureau takes first step to revising payday lending rule | Trump wants to loosen rules on bank loans | Pentagon, FDA to speed up military drug approvals MORE (R-Maine) have been working with Republicans who opposed the measure, such as Sens. John McCainJohn Sidney McCainMcCain rips Trump for attacks on press NSA spying program overcomes key Senate hurdle Meghan McCain says her father regrets opposition to MLK Day MORE (R-Ariz.) and Kay Bailey Hutchison (R-Texas), to resolve some issues.

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On Tuesday, the Senate approved a motion to proceed to the Sportsmen’s Act, S. 3525, which would increase access to federal land for hunters and fishermen while also supporting conservation measures. Amendments to that bill are due by 4 p.m.

Reid also said he was optimistic that Republicans and Democrats would agree to a list of amendments to advance a vote on the sportsmen's bill by the end of Wednesday, rather than Thursday. 

Reid said he hopes to have an agreement on both bills "in the next couple hours."

A Senate aide said if a deal is reached, a vote on the cybersecurity bill could happen as early as Wednesday.

Reid voted against the bill as a procedural move in August so he could bring it back to the floor at a later time. Because of this, the Senate aide added that Reid can bring up the cybersecurity bill at any time as a motion to reconsider that vote.

The next bill the Senate will take up is the Defense Authorization Act, S. 3254.

Jennifer Martinez contributed.