The Kerry-Graham-Lieberman proposal was slated to surface April 26, but the rollout was scuttled when Graham suspended his support for advancing the measure.
Graham is furious that Democratic leaders plan to push immigration legislation this year, claiming it will spark a divisive political battle that will sink prospects for the climate measure.
Lieberman signaled this week that he and Kerry may unveil the measure without Graham, while also stressing that Graham supports the substance of the bill. Lieberman told The Hill Tuesday that the idea of moving ahead without Graham is an "open question."
"We are talking to him all the time. . . . He is still supportive of the bill, but because of all the political crosscurrents about immigration reform, he is not yet, as I understand it, prepared to join us in launching it," Lieberman said.
Kerry on Wednesday declined to directly address the possibility of unveiling the bill without Graham, telling reporters that Graham is "standing by the work product and he is standing by the bill, no matter what."
Reid said last week that be plans to bring up energy and climate legislation before an immigration bill, noting it is further along.