The Vermont Democrat, third-in-line on the panel, strongly denied those claims. But all four sources said Leahy has sent several signals to Majority Leader Harry Reid (D-Nev.) about the move.

Leahy “has tried to meet with Reid several times” about Byrd’s chairmanship, according to another source.

The 90-year-old Byrd has been ailing for months, after having been hospitalized in February after a fall in his home and again in March for adverse reactions to medication. Both hospital stays were short, but senators this week raised the issue of his fitness as chairman during a Democratic leadership meeting.

Although speculation intensified this week, informal chatter has been circulating for months about Byrd’s abilities and workload, several sources noted. Other candidates mentioned for the chairmanship include Democrats Patty Murray of Washington, who has already helped handle some of Byrd’s workload. Murray strongly denied any interest in Byrd’s position when asked Thursday.

Traditional succession on the Appropriations Committee favors Sen. Daniel Inouye (D-Hawaii), who already chairs the Senate’s Appropriations Defense Subcommittee, before Leahy.

Last month, Inouye noted that he was "next in line" but said he didn’t even “want to think about” succeeding Byrd.

Yet, there is little evidence to suggest Inouye is actively pursuing the post. An Inouye spokesperson did not return an immediate call for comment.

Leahy, who chairs the Senate Judiciary Committee, said he is not pushing for the position publicly or privately and that his real interest remains in the Judiciary Committee.

“Oh hell no. Good God, no. I don’t know where that rumor started,” Leahy said. “I will support the seniority system. I will support Senator Byrd as long as he wants to be chairman. I will vote for whoever is next in line.”

Leahy said he has not met with Reid on the issue, "other than general discussion that has been going on on other things."

Following a meeting at the White House on Wednesday, Reid told The Hill that he talked to Byrd Tuesday night and that the senator’s chairmanship would “continue until something else comes up.”

One Democratic senator who spoke on condition of anonymity said Thursday that Leahy has been pushing for the appropriations gavel “a little.”

Byrd’s spokesman, Jesse Jacobs, on Thursday dismissed the speculation and said the senator remains engaged and in charge.

“There are 100 rumors a day around here,” Jacobs said.

The jockeying comes at a key time in the appropriations process. Appropriations subcommittees are providing recommendations to committee leaders on the $100 billion-plus wartime supplemental spending bill that will hit the Senate floor later this spring.  

Democrats are questioning whether Byrd's health might prevent him from taking the lead on what will likely be a bitter fight over the measure.

Most Democrats are publicly hesitant to discuss the possibility of replacing Byrd, who has long been viewed as the Senate’s premier statesman.

"I'm proud to serve with Robert Byrd and I support Robert Byrd," said Sen. Byron Dorgan (D-N.D.).

Manu Raju contributed to this article.