Bayh, who has spoken with Senate Majority Leader Harry Reid (D-Nev.) about his initiative, said he is trying to create a faction of moderate Senators who will gather on a weekly basis ahead of the usual Tuesday Democratic Caucus meetings.

Additionally, Bayh envisions inviting outside speakers to address the group, which would also work in concert with third parties that have similar viewpoints, like the Third Way, a nonpartisan progressive think tank.

A...