Democratic Sen. Jay Rockefeller says he's "feeling a bit better" about Barack Obama's pick for CIA chief, after he and other senators initially voiced discontent with the president-elect's choice. 

Rockefeller, D-W.Va., the outgoing chairman of the Senate intelligence committee was concerned that he was not consulted on Obama's surprise pick, former White House chief of staff Leon Panetta. Incoming committee chairwoman Dianne Feinstein, D-Calif., had similar concerns. 

But Feinstein dropped her criticism and backed Panetta after Obama and Vice President-elect Joe Biden apologized for not consulting her. 

Rockefeller, too, told FOX News he's now considering the selection. Rockefeller is set to meet next week with Panetta. 

The meeting comes after Rockefeller spoke with his top choice for the job, CIA deputy director Steven Kappes. He said Kappes is also meeting with Obama's expected nominee and "wants to feel comfortable with (Panetta), a man he has never met." 

Kappes was also Feinstein's original choice, and both senators want to see him stay as Panetta's No. 2. 

"I think there is a very good chance he will be there," Rockefeller said. 

FOX News' Trish Turner contributed to this report.