WASHINGTON — President Trump said on Wednesday that he intended to name Kirstjen Nielsen, a top White House aide, to lead the Department of Homeland Security, elevating a former homeland security official in the George W. Bush administration who has lately worked to impose order in Mr. Trump’s chaotic West Wing.

Mr. Trump announced his choice in a statement that noted Ms. Nielsen’s “extensive professional experience in the areas of homeland security policy and strategy, cybersecurity, critical infrastructure and emergency management.” She is the first nominee for the homeland security post who had served in the department, according to the statement.

If confirmed, Ms. Nielsen would replace John F. Kelly, who was homeland security secretary until he left in July to serve as the White House chief of staff and bring discipline and direction to a West Wing plagued by disorganization and infighting. Mr. Kelly had drafted Ms. Nielsen to be his chief of staff at the Homeland Security department, and when the president plucked him for the White House, he brought her as his No. 2.

Known as a no-nonsense player and policy wonk, Ms. Nielsen appears unlikely to land at the center of the type of controversies that have engulfed Mr. Trump’s presidency. But her regimented style in a freewheeling and often dysfunctional West Wing frustrated some senior officials and people close to the president, who chafed under her dictates. On Wednesday, some of them described Ms. Nielsen’s promotion as a solution to a toxic personnel situation, while others fretted privately that her departure would create a void at the White House that would be difficult to fill.