The group includes moderates and senators known for crossing party lines, a few who are retiring and some who had already broken with their party to support same-sex marriage.
Only two of the 14 Republicans who broke ranks to support taking up gun legislation are facing voters this year, showing how difficult such deals could be in the future.
Senator Lindsey Graham voted last year to confirm Judge Ketanji Brown Jackson to an influential appeals court. He now calls her an “activist judge” who should not ascend to the Supreme Court.