New Orleans, Louisiana - A Texas law that would allow state police to arrest and deport migrants they suspect of crossing into the United States from Mexico without documentation was again placed on hold late Tuesday, the latest in an ongoing legal back-and-forth over its fate.

The administration of Democratic President Joe Biden has strongly opposed the law, known as Senate Bill 4, arguing that the federal government has authority over immigration matters, not individual states.

"SB 4 will not only make communities in Texas less safe, it will also burden law enforcement, and sow chaos and confusion at our southern border," White House Press Secretary Karine Jean-Pierre said in a statement. "SB 4 is just another example of Republican officials politicizing the border while blocking real solutions."

A federal judge last month temporarily blocked the law passed by the Republican majority in the Texas state legislature saying it "conflicts with key provisions of federal immigration law."

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But a conservative-dominated appeals court said SB 4 could go into force unless the Supreme Court ruled otherwise.

The nation's highest court, which has a 6-3 conservative majority, issued a temporary stay on SB 4 earlier this month but lifted it Tuesday, allowing it to take effect while legal challenges play out in lower courts.

But by Tuesday evening, the 5th Circuit Court of Appeals put the law back on hold. That hold, however, could be soon again reversed – allowing the law to go back into effect as arguments over it continue.

The three liberal justices on the Supreme Court had dissented.

"Today, the Court invites further chaos and crisis in immigration enforcement," Justice Sonia Sotomayor wrote.

"Texas passed a law that directly regulates the entry and removal of noncitizens and explicitly instructs its state courts to disregard any ongoing federal immigration proceedings," Sotomayor said. "That law upends the federal-state balance of power that has existed for over a century."

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Mexico said Tuesday it "will not accept, under any circumstances, repatriations by the state of Texas," including Mexican citizens.

Migrants in Mexico told AFP they still planned on crossing the border.

"We've come to work," said 42-year-old Oscar Galeano of Guatemala, hoping for compassion from US authorities and an opportunity to immigrate. "We didn't come to take anything from anyone."

Venezuelan Giancarlo Navarro (43) described himself as "a political exile."

"I cannot return to my country," he said.

Republicans blame Biden for the recent record arrivals of migrants into the United States. Meanwhile, the White House accuses Republicans of deliberately sabotaging a bill it supports which would enact some of the toughest border policy changes in decades and lead to further militarization of the region.

Greg Abbott, the Republican governor of Texas and an ally of Republican presidential candidate Donald Trump, has decried an "invasion" of the southern border.

"Texas has the right to defend itself because of President Biden's ongoing failure to fulfill his duty to protect our state from the invasion at our southern border," Abbott said recently.

SB 4 is the latest immigration flashpoint between Abbott and federal authorities.

The US Justice Department has also filed a lawsuit seeking the removal of a floating barrier installed by Texas authorities in the Rio Grande river to stop migrants crossing from Mexico.