Wolf reportedly helped come up with the Trump administration's "zero tolerance" policy that separated migrant families at the border.
Administration officials are reportedly exploring a loophole that would allow Trump to bypass a federal law to appoint Ken Cuccinelli or another favored pick.
Kevin McAleenan will testify Wednesday before a House committee on global terrorism.
McAleenan struggled to mesh with the Trump administration.
Activists said the acting secretary's appearance at a law school conference would have given the Trump administration a "platform to spread hatred."
Kevin McAleenan expressed frustration to The Washington Post about his inability to control the department's approach "in an increasingly polarized time."
U.S. immigration officers arrested 680 people just a few days after a gunman in El Paso, Texas, targeted Mexicans in a mass shooting.
Communities scrambled to find beds and food for children suddenly left alone at schools and day care after their parents were rounded up in mass raids.
Freshman lawmaker Alexandria Ocasio-Cortez asked acting DHS Secretary Kevin McAleenan if Trump's policies had contributed to a "dehumanizing culture."
Homeland Security officials linked the decline to President Donald Trump's hard-line immigration policies and help from Mexico.