President Donald Trump’s campaign to oust Federal Reserve Governor Lisa Cook, if successful, would give him the opportunity to exert more influence over the US central bank by securing a majority on its seven-member board of governors.

Mortgage fraud has become a big focus for the Trump administration.

Bill Pulte, director of the agency that oversees mortgage giants Fannie Mae and Freddie Mac, urged the Justice Department to investigate Cook, who was appointed to the Fed’s governing board by former ...