Installing Mr. Conway to lead the civil division means that defending the president from such challenges will become a family affair for the Conways. Ms. Conway, a staunch loyalist who ran the final months of Mr. Trump’s presidential campaign, has been a frequent presence on television news programs promoting the president’s agenda and dismissing criticism of his style and record.

Her zeal on Mr. Trump’s behalf has sometimes landed her at the center of controversy, such as when she claimed that the White House was entitled to put forward “alternative facts” about the crowd size at his inauguration, and in a separate interview a few weeks later, referred to a terrorist attack in Bowling Green that never occurred. Last week, she appeared to suggest that President Barack Obama might have spied on Mr. Trump through a microwave. Ms. Conway later clarified that she was speaking in general about possible means of surveillance, not about Mr. Obama, and Sean Spicer, the White House press secretary, said she had been joking.

Mr. Conway had been a contender for the job of solicitor general for the Trump administration, but Mr. Trump announced this month that the job would go to Noel J. Francisco.

Mr. Conway is a partner at the New York City firm of Wachtell, Lipton, Rosen & Katz. He specializes in securities, contract and antitrust litigation, as well as mergers and acquisitions, according to his biography on the firm’s website. He is a graduate of Harvard University and the Yale Law School.

While there is a law against nepotism in government, it would not affect the Conways. It says that no public official can hire a family member — including one related by marriage — to serve in an agency or office over which he or she has authority. Ms. Conway would have no direct authority over her husband were he to be confirmed, nor would the reverse be true.