That those past comments may hinder her chances are a reminder that Mr. Trump does not have a large circle of loyalists to choose from for top positions.

Other candidates are also being considered, including David Malpass, the under secretary of the Treasury for international affairs, whose loyalty to the president runs deep: He served as an economic adviser on Mr. Trump’s 2016 campaign. Another name in the mix is Ray Washburne, the president of the Overseas Private Investment Corporation. Mr. Washburne also served on Mr. Trump’s 2016 finance team and was seen as an early candidate for a cabinet position.

The World Bank’s current president, Jim Yong Kim, said this month that he would step down in February to join a private infrastructure investment firm, an unexpected departure that comes nearly three years before the end of his term. Mr. Kim was first appointed by the Obama administration in 2012, on the recommendation of Hillary Clinton when she was secretary of state, and was reappointed by President Barack Obama in 2016 to serve a second five-year term.

The process of choosing a successor is being overseen by Steven Mnuchin, the Treasury secretary; Mick Mulvaney, the acting chief of staff; and Ms. Trump, whose role in the process was announced by the White House on Monday.

Ms. Trump’s role in the process drew some criticism from ethics watchdogs on Monday, who said that it could pose a conflict of interest for the president’s daughter to be involved in international economic matters when she has not completely divested from her assets.

But Ms. Trump’s role is notable because of her unique sway over her father when it comes to personnel matters. The group was expected to begin the interview process on Tuesday in order to present Mr. Trump with recommendations for a nominee.

A White House spokesman did not respond to a request for comment about the candidates.