Minutes after Brett Kavanaugh was confirmed to the U.S. Supreme Court, Kannon Shanmugam took to Twitter to publicly congratulate the new justice.

Shanmugam's tweet highlighted a July post in which the Williams & Connolly appellate partner congratulated Kavanaugh on his nomination to the high court. He gave similar congratulations to Justice Neil Gorsuch following his confirmation last year.

Shanmugam, with at least three cases at the Supreme Court this term, has no shortage of business coming before Kavanaugh—unless the former clerk to Antonin Scalia gets a new gig.

Kavanaugh's elevation to the Supreme Court—he takes the bench Tuesday as a justice—created a new prize: an opening on the U.S. Court of Appeals for the D.C. Circuit, where Kavanaugh had served since 2006.

Just days after the Senate's 50-48 vote Oct. 6 to confirm Kavanaugh, Washington's appellate bar was abuzz about who might be nominated to fill the newly vacant seat on the D.C. Circuit. Shanmugam's name, along with several others, were making the rounds Monday on social media and in interviews with lawyers.

“You could fill a dictionary with the number of lawyers who would want to be on the D.C. Circuit—and actually believe they could be on the D.C. Circuit,” said George Mason University's Jamil Jaffer, a former clerk to Justice Neil Gorsuch and George W. Bush White House lawyer.

The White House was mum about any D.C. Circuit short lists, and none of the lawyers speaking about the vacancy purported to have any inside information about whom the Trump administration might pick, and when.

The names of several former clerks to Justice Clarence Thomas—himself a former D.C. Circuit judge—were making rounds Monday as would-be contenders. They included Quinn Emanuel Urquhart & Sullivan partner Christopher Landau and Neomi Rao, who has played a leading role implementing Trump's deregulatory agenda as administrator of the Office of Information and Regulatory Affairs.

Rao came to the administration from George Mason University's Antonin Scalia Law School, where she was the founding director of the Center for the Study of the Administrative State. Landau is a leading Supreme Court advocate who'd spent years at Kirkland & Ellis before moving to Quinn Emanuel.

Rao could not be reached for comment, and Landau declined to comment. Shanmugam did not return a message seeking comment.

Six of Trump's 26 federal appellate appointees are women, and he could be compelled to pick a female lawyer to fill Kavanaugh's seat, following his terse confirmation proceedings that involved allegations of sexual misconduct. Kavanaugh denied claims from California research psychologist Christine Blasey Ford that he sexually assaulted her at a house party in Maryland in the 1980s.

Another possible contender within the Trump administration is Hashim Mooppan, a Justice Department appellate lawyer and former Jones Day partner who earlier worked closely with White House Counsel Donald McGahn.

Mooppan, a former clerk to Antonin Scalia, has argued several prominent cases in federal appeals courts defending Trump administration policies. In the Second Circuit, he argued federal civil rights laws do not protect against sexual orientation discrimination, and he argued against the constitutionality of the Consumer Financial Protection Bureau's single-director structure.

Kate Todd, a former senior vice president and chief counsel of the U.S. Chamber of Commerce's Litigation Center, is also seen as a possible contender for the D.C. Circuit. She did not respond to a message seeking comment. Kirkland & Ellis partner Erin Murphy, a former clerk for Chief Justice John Roberts Jr., was also floated in one conversation as a possible pick. She could not be reached for comment Monday.

Williams & Connolly is already on Trump's nominations radar. The administration recently nominated one of Shanmugam's colleagues—Allison Jones Rushing—to the Fourth Circuit. The 2007 Duke Law School graduate would be one of the youngest judges confirmed to the federal bench in years. Former Williams & Connolly partner Emmet Flood is on the team defending the president in the special counsel's Russia investigation.

Another lawyer in Washington suggested the White House could turn to Carl Nichols, a Wilmer Cutler Pickering Hale and Dorr government and regulatory litigation partner. Nichols is already a nominee to Washington's federal trial court. The former Thomas clerk has a long appellate background at the U.S. Justice Department. Nichols declined to comment Monday.

Trump's only made one pick so far for the D.C. Circuit: Gregory Katsas, a former Jones Day partner who joined the appeals bench from the White House counsel's office. Katsas was a Thomas clerk during the term following his 1991 confirmation proceedings.

Read more: