By Geneva Sands, CNN

(CNN) -- The Senate is moving ahead with Chad Wolf's nomination for Secretary of Homeland Security amid concerns about the legitimacy of his appointment, with a full committee hearing scheduled for Thursday morning.

Wolf, who has served for 10 months as acting Homeland Security secretary, was formally nominated by President Donald Trump on September 10.

The nomination hearing comes as courts and the Government Accountability Office have raised concerns about the legitimacy of Wolf's appointment to lead DHS. Over the objections of the department, the GAO stood by its August opinion that Wolf and his deputy Ken Cuccinelli were appointed as part of an "invalid order of succession."

On Wednesday he will be introduced by Sen. Ted Cruz of Texas at the Senate Homeland Security and Governmental Affairs Committee hearing.

DHS hasn't had a Senate-confirmed leader since Secretary Kirstjen Nielsen's forced departure in April 2019. Trump has repeatedly lavished praise on Wolf. "Thank you very much, Chad. You're right in the heart of it, and you've got some big things coming," Trump said at an August event. "Good job."

Last week, Wolf was a no-show at a House hearing on threats to the US homeland after a feud between the department and the committee over scheduling. DHS cited Wolf's nomination as reason for not attending the House panel hearing. Instead, Wolf participated in a meeting with Senate staff regarding his nomination to be secretary.

The move angered Democratic Chairman Bennie Thompson of Mississippi, who denounced Wolf's decision to defy a subpoena to testify before the House Homeland Security Committee.

"As the person running the Department of Homeland Security, Mr. Wolf should be here to testify, as secretaries of Homeland Security have done before. Instead, we have an empty chair -- an appropriate metaphor for the Trump administration's dereliction on so many of these critical homeland security issues," Thompson said at the hearing.

Despite the upcoming hearing, it's unclear whether Wolf will get a Senate vote before the November presidential election.

Senate Minority Leader Chuck Schumer wrote of Wolf that "given his past actions, he'd be an awful choice."

When Trump announced Wolf's nomination, the committee's Democratic ranking member Sen. Gary Peters of Michigan said in a statement that given Wolf's record since assuming his role, he has "serious questions and concerns about his suitability for the job," saying the department "needs qualified, principled leaders to safeguard" the nation.

Peters will carefully consider any nomination that comes before the committee, according to a Peters' aide. The ranking member previously opposed Wolf's nomination to serve as DHS Under Secretary for Strategy, Policy, and Plans.

Wolf grew up in Mississippi and Texas and worked for a lobbying firm for more than 10 years before returning to the federal government at the start of the Trump administration.