Washington (CNN) John Kelly's elevation to White House chief of staff has left an opening at the top of the Department of Homeland Security, and names are already circulating throughout Washington for his possible replacement.

It remains unclear how seriously any of those individuals are being considered.

Since Kelly took his position as President Donald Trump's right-hand man, he is expected to have an outsized role in the process of finding his replacement.

"I think he will have a great deal of say in who goes over to replace him because he has had this position," a source close to Kelly said. "He now has a very soft spot in his heart for the Department of Homeland Security," the source added, saying that it's "now his baby."

Kelly is especially concerned about his replacement continuing the trajectory he feels he got the department on, particularly on issues like morale of the workforce, the source said.

Multiple sources close to the White House and the administration expect the process will take some time, and that having acting DHS Secretary Elaine Duke, who was confirmed this spring as the deputy secretary, reduces the pressure to speed things up.

"Elaine Duke is a steady hand at the wheel so you don't need to rush this process, because she's very, very steady," said a congressional aide. "She can run the ship well, so they can take their time."

Here's a look at some of the options being discussed.

Jeff Sessions

Given Trump's public berating of Sessions in recent weeks, there was speculation that the President could move Sessions to the opening at DHS to get him out of the Justice Department without firing him.

Doing so would eliminate what was a growing source of frustration for the President -- namely Sessions' recusal from the investigation into Russian election meddling and the special counsel at DOJ -- and puts an experienced hand atop an agency key to Trump's immigration and counter-terrorism agenda, all without firing a man who was defended aggressively by members of Congress and conservative organizations.

According to the Federal Vacancies Act, Trump is able to replace anyone who is confirmed by the Senate to hold the open position, but if he chooses Sessions, Trump would be limited from selecting him as a permanent replacement. Sessions could serve up to 210 days, according to a Congressional Research Service report

But the White House has denied that option is under consideration.

"There are no conversations about any Cabinet members moving in any capacity, and the President has 100% confidence in all members of his Cabinet," White House press secretary Sarah Sanders said when asked about the rumor on Monday.

Mike McCaul

The Texas congressman, who's also the chairman of the House homeland security committee, is considered a strong contender for the position given his experience and border-state ties. He also would likely face an easy road to confirmation among his peers in the upper chamber.

He also has had a good relationship with Kelly during his tenure, the aide said.

The congressional aide said McCaul would happily take the job. But two sources familiar with discussions about the matter cautioned CNN that McCaul has concerns about taking the position due, in part, to how Trump has treated other top administration officials. One the sources said Trump's recent criticism of Sessions and other episodes of White House chaos are giving McCaul "pause."

"How the administration has leapt from one chaotic episode to the next and seeing how Trump loyalists like Sessions and (New Jersey Gov. Chris) Christie have been treated by the President will likely cause pause," the well-placed source said.

And like during the initial process of selecting Trump's secretary of homeland security last November, McCaul faces fierce opposition from groups advocating for the restriction of immigration and hard-line enforcement of immigration laws.

NumbersUSA, which lobbies and mobilizes for strict immigration laws, put out a statement of opposition just at the suggestion of McCaul's name.

"Congressman McCaul's record in the House suggests that he doesn't have the inclination or experience to continue and improve on the great start that General Kelly has had in leading DHS to be a pro-worker institution," said NumbersUSA President Roy Beck. "His actions in recent years put him in the bottom third of Republican representatives in terms of immigration enforcement and policies that achieve Pres. Trump's priority of putting American workers first."

Elaine Duke

Duke's time serving as acting secretary could end up serving as an audition of sorts, with former DHS officials singing her praises as a longtime civil servant.

Duke has a nearly three-decade career in government, the last eight of which were spent at DHS. She has expertise in acquisition and management. Duke was confirmed as deputy secretary in April on an overwhelmingly bipartisan vote of 85 to 14.

James Norton, a former DHS official in President George W. Bush's administration who worked with Duke, said her experience with acquisition and reorganizing DHS' emergency management after Hurricane Katrina make her a strong candidate.

"One of the biggest struggles for DHS is acquisition and to have someone at the top that understands that process is good," Norton said. "From the standpoint of a new cybersecurity organization (being debated in Congress), which would be the biggest reorganization of DHS since Katrina reform, she's well suited to that."

Norton also noted that her acting stint makes her one of the highest ranked women in the Cabinet, especially in the national security space.

"I think it's an opportunity to perform on a big stage," he said. Another former DHS official calls her a "bureaucrat's bureaucrat" who "doesn't make waves" and is respected on both sides of the aisle.

Kris Kobach

The Kansas secretary of state has been a prominent advocate of aggressive immigration policies at the state level, having helped write Arizona's controversial immigration law -- and has long been supported by the immigration hawks for a role in the Trump administration.

But another person close to Kelly called Kobach a "non-starter," and he is not seen as confirmable in the Senate, according to the congressional source. Kobach does have a role in the administration, as co-chairman of Trump's commission on voting integrity, and that tenure alone has brought controversy and court challenges.

The White House did not rule out Kobach as a possibility, but asked on Tuesday about the suggestion, Sanders indicated it wasn't in the works.

"I'm not aware of any movements for him," Sanders said, adding the White House would share personnel announcements as they come up.

Tom Cotton

The Arkansas senator is a national security hawk, a war veteran and has authored a bill favored by immigration restriction groups to cut back on non-skills-based immigration to the US. The White House has been working with Cotton and his co-author on the bill for possible re-introduction.

Cotton does not have a long history of working on DHS-related issues outside of that, however.

Asked by CNN on Tuesday at the Capitol if he was being considered, Cotton ignored the question and did not offer any denial.

Thomas Homan

The acting director of US Immigration and Customs Enforcement has established himself as a favorite of Trump's, accompanying him on Air Force One last week to speak at an event on Long Island before law enforcement on the administration's anti-gang policy.

Trump praised Homan at that event, saying he has "done an incredible job."

"He's a tough guy. He's a tough cookie," Trump said. "Somebody said the other day, they saw him on television, and somebody -- they were interviewed after that -- they said, he looks very nasty, he looks very mean. I said, 'That's what I'm looking for.'"

Homan is a long-time law enforcement officer, having worked as a police officer, on the border and on the ground in immigration enforcement. A former DHS official who worked with Homan suggested that Trump likes their New York connection and that he has served on the front lines.

But there is also concern about whether Homan would be able to be confirmed in the Senate. Homan has served as the face of Trump's harshest immigration policies, saying undocumented immigrants "should be afraid" of law enforcement under this administration, which may be difficult for purple-state Republicans to get behind. Homan would also face tough questioning about the future of the Deferred Action for Childhood Arrivals program from Republicans like South Carolina Sen. Lindsey Graham, who support the program.

Homan has not been formally nominated for his position atop ICE and thus has not faced the Senate confirmation process.

One person close to Kelly said: "Trump loves the guy -- (but he) has some limitations as a manager."

James Loy

One of the people close to Kelly also suggested that Loy could be in the mix. Like Kelly, Loy is a veteran of the military, having served as commandant of the US Coast Guard.

Loy also was deputy secretary of DHS from 2003 to 2005 and was the administrator of the Transportation Security Administration before that.

He retired from government in 2005 and went into the private sector.

Loy did not respond to a request for comment as to whether he'd consider going back into government.

Rick Perry

The person close to Kelly also said Trump's energy secretary could be in the mix.

Perry was governor of Texas, a position that put him on the front lines of border security. In 2014, he deployed the Texas National Guard to the border to help with a surge of undocumented immigrants.

Asked about the possibility of a job switch for Perry, Shaylyn Hynes, a Department of Energy spokesperson, said, "While Secretary Perry is honored to be mentioned, he is happy where he is and very focused on carrying out the mission the President gave him when he was chosen lead the Department of Energy."

Rudy Giuliani

The former mayor of New York has been a perennial shortlist candidate for running the department going back to the Bush administration, but has never been formally nominated for the job. He gained nationwide prominence during the aftermath of the 9/11 attack and has built a career in the private sector around his expertise.