Wolf will become the fifth person to serve in the role since the beginning of the Trump presidency.
When a reporter asked whether or not a Sharpie was used to alter the map, Trump replied, "I don't know, I don't know."
"We have a multi-mission focus always at the Department of Homeland Security," said secretary Kevin McAleenan.
"The situation is improving by every available metric, but, I want to be very clear that we remain at and beyond crisis levels in illegal crossings," said Acting Secretary of Homeland Security Kevin McAleenan.
Kevin McAleenan's remarks come days after a gunman killed at least 22 people in a racist attack at a Texas Walmart.
Acting DHS Secretary Kevin McAleenan pleaded for more funding at the border, but Michigan congresswoman Rashida Tlaib called it a "broken" system.
Kevin McAleenan praised Donald Trump for playing hardball with the Mexico tariff threats, says he forced them to make "real proposals."
"It's intentional. It's a policy choice being made by this administration, and it's cruel and inhumane," said Representative Lauren Underwood, an Illinois Democrat.
Grady reportedly has no intention of stepping aside for Customs and Border Protection Commissioner Kevin McAleenan to become acting secretary.
"It's been more than a decade that we've had a child pass away anywhere in a CBP process so this is just devastating for us," Customs and Border Patrol commissioner Kevin McAleenan told "CBS This Morning" on Wednesday.