Under a change to the rules, which has been openly considered by lawmakers in the chamber, Senate Democrats would use a majority vote to prevent Republicans from being able to filibuster nominees to the courts and federal agencies.
Republicans claim that Senate Majority Leader Harry Reid (D-Nev.) promised not to trigger the option during negotiations over rules reform at the beginning of the year. Democrats, however, say Republicans have not lived up to their end of the bargain.
As precedent, Cohen cited then-Senate Majority Leader Bill Frist (R-Tenn.), who threatened to change the body's rules in 2005 before Democrats, then in the minority, agreed to allow an up or down vote on a slate of President George W. Bush's judicial nominees.
The term for NLRB Chairman Mark Gaston Pearce is scheduled to end in August, bringing urgency to the decision on whether or not to change the rules.
"The Senate majority must act in July," Cohen said.
The legal authority of the NLRB has been muddied by a January opinion from the D.C. Circuit Court of Appeals that found that two current members to the board were unconstitutionally appointed by President Obama. The decision, which the administration has appealed to the Supreme Court, would mean that the NLRB was not able to form a quorum, potentially nullifying all of its recent actions.
The NLRB is a five member board, but there are currently two vacancies.
A Senate committee voted along party lines in May to send President Obama's slate of five nominees to the full chamber.