Cruz said on Friday that the USA was facing a "humanitarian and security crisis at our southern border", and warned that the influx of migrants could bring in new cases of the deadly coronavirus strain.
Media reports have said migrants will be tested for the coronavirus in Mexico before entering the U.S.
"I am very happy because in all this time we have been here waiting we have had many unpleasant experiences", she said.
"Honestly, I have no words for how I'm feeling right now!"
Biden vowed on the campaign trail to roll back restrictive Trump-era immigration policies, including MPP, which is informally known as "remain in Mexico".
"Last night, I was in the phone for two straight hours with Xi Jinping", Biden told reporters - an unusually long interaction for a United States president, with whom even face-to-face meetings rarely stretch beyond an hour.
The Biden administration has continued to gut President Trump's border policies amid a major increase in illegal border crossings.
On President Joe Biden's first day in office, the U.S. Department of Homeland Security suspended the "Remain in Mexico" policy for newly arriving immigrants seeking political asylum.
In Mexico, President Andres Manuel Lopez Obrador, at his press briefing, welcomed Biden's move.
"Building on a series of Executive Orders last week, the Biden Administration is announcing another step in our phased strategy to reform the nation's immigration system", the Department of Homeland Security announced on Thursday in a statement.
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The officials, all of whom requested anonymity, only outlined procedures for people with active cases in United States immigration courts.
It said there are approximately 25,000 active cases still.
Under the plan, migrants will need to register with worldwide organizations over the internet or by phone and await instructions.
The United States reported almost 3,900 new deaths from COVID-19 Thursday, pushing the USA death toll past 475,000, by far the highest in the world.
The migrants' claims will initially be heard at three ports of entry along the border, with organizations working to identify those in line the longest and those deemed particularly vulnerable. They plan to start slowly, with two border crossings each processing up to 300 people a day and a third crossing taking fewer numbers.
Administration officials said Washington would continue to apply targeted sanctions against Chinese individuals and entities for intellectual property theft, and tighten China's access to sensitive technologies.
The Biden officials declined to name the ports where the administration planned to begin processing, citing concerns that people would rush to those locations.
"We're looking for every opportunity we can, of course, from the president to the secretary of Homeland Security to the secretary of State to communicate where we stand", the press secretary continued. The pandemic-era order allows U.S. authorities to rapidly expel to Mexico migrants caught crossing the border illegally, a practice advocates say bypasses due process.
The Biden administration has not said whether it will end that program. "As more migrants catch wind of the reimplementation of 'catch-and-release, ' the surge on our border will be unimaginable".
The effort to process MPP enrollees comes as arrests of migrants at the US-Mexico border have been increasing after a steep drop at the beginning of the coronavirus pandemic.