Deputy Attorney General Rod Rosenstein, who confidants said expected to be fired Monday, will meet face-to-face with President Trump on Thursday to discuss whether he keeps his job as the No. 2 official in the Justice Department.

Mr. Rosenstein, who oversees the special counsel investigation into alleged Russian interference in the 2016 election, has frequently been the target of Mr. Trump’s ire. But his position appeared more tenuous after the New York Times reported Friday that in early 2017 he discussed secretly recording...