Videos and pictures showing a border guard apparently threatening migrants with the cords were shared on social media.
"I don't think anyone seeing that footage would think it was acceptable or appropriate," White House spokeswoman Jen Psaki told reporters on Monday.
"I don't have the full context. I can't imagine what context would make that appropriate," she added.
Some on social media commented that the image of fleeing black men chased by white officers on horseback had echoes of the historical injustices suffered by black people in the United States.
US Border Patrol Chief Raul Ortiz said the incident was being investigated to make sure that there was not an "unacceptable" response by law enforcement. He said officers were operating in a difficult environment, trying to ensure the safety of the migrants while searching for potential smugglers.
Department of Homeland Security Secretary Alejandro Mayorkas said the long reins are used by mounted officials to "ensure control of the horse."
"But we are going to investigate the facts," he said during a news conference in Del Rio.
The camp under a bridge spanning the Rio Grande has become the latest flashpoint for US authorities seeking to stem a flow of migrants fleeing gang violence, extreme poverty and natural disasters in their home countries.
US border agents filmed swinging rope at migrants
By Monday, hundreds of migrants had returned to the Mexico side amid uncertainty about whether they would be deported back to Haiti on flights organised by US authorities.
The first flights carrying migrants landed in Port-au-Prince on Sunday from the Del Rio camp arrived in Haiti on Sunday, with at least three more due to make the journey on Monday, according to flight-tracking website Flightaware.
US officials closed the Del Rio border crossing last Friday due to the crush of migrants, and said Monday it remained shuttered, with most traffic re-routed to the Eagle Pass, Texas, border crossing, some 90 km south.