Likely Democratic targets include not only the president’s personal finances and those of the Trump Organization, but also the actions taken by his sons Donald Jr. and Eric and his son-in-law, Jared Kushner, during the 2016 campaign and its aftermath.

The Oversight and Judiciary Committees are likely to focus on any violations of the Constitution’s emoluments clause, which bans payments from foreign governments to federal officeholders, an avenue that will lead to an examination of payments to Trump-held properties and the role of both of his sons and Mr. Kushner. And some investigators may try to question Donald Trump Jr. again about a meeting he held with a Kremlin-linked lawyer promising “dirt” about Mrs. Clinton.

While some of the areas primed for inquiry — most notably, ties between the Trump campaign, the Trump Organization and Russia, and possible campaign finance violations — are likely to overlap with investigations by the special counsel, Robert S. Mueller III, and federal prosecutors in New York, Democrats believe they have a broader mandate to question everything from foreign business dealings by Ms. Trump and the Trump Organization to the administration of security clearances at the White House, including the one given to Mr. Kushner.