Sen. Patty Murray of Washington, the No. 3 Senate Democrat, said in a statement last week that “there needs to be a discussion” on the issue. Durbin’s allies, however, are quick to highlight that Murray is both in a high-ranking leadership position and is the top Democrat on the Senate HELP Committee. The issue is expected to come up during Democrats’ weekly caucus call Tuesday and has already been brought up in smaller meetings.

The talks could lead to a caucus vote on their rules in the coming days. Unlike Senate Republicans, Democrats do not have term limits on leadership positions, chairmanships and ranking member roles.

Durbin and Whitehouse have taken different approaches to the race. While Durbin has publicly announced he wants to lead Judiciary, Whitehouse has simply stated that the caucus should decide and he will abide by their choice.

Durbin supporters say that the party’s chief vote-counter already has the support to secure the Judiciary post and keep his job as whip. But several progressive groups are pushing for Whitehouse, who they view as a more aggressive combatant on the committee.