According to sources familiar with the matter, President Joe Biden is expected to sign an executive order addressing immigration issues as early as this week. The long-awaited order aims to limit the number of migrants who can claim asylum at the southern U.S. border.

Under the executive order, migrants seeking asylum at the border would be immediately sent back to Mexico, where they would have to wait until the daily average of asylum claims decreases. Once the number of claims goes down, they would be allowed to proceed with their asylum applications. The exact threshold that would trigger a pause on claiming asylum is still being discussed.

Sources have revealed that members of Congress have already been briefed on the upcoming executive action. However, administration officials caution that any such order would likely face legal challenges.

"I anticipate that if the president would take executive action, and whatever that executive action would entail, it will be challenged in the court," said Homeland Security Secretary Alejandro Mayorkas during a press conference last month.

The Biden administration has been urging Congress to pass a bipartisan border bill that was proposed earlier this year, in addition to considering executive action. The bill is aimed at addressing the challenges faced by the immigration system.

In light of these developments, the White House has invited Brownsville, Texas Mayor John Cowen and El Paso Mayor Oscar Leeser to attend a meeting on Tuesday for an immigration-related announcement. Mayor Leeser expressed gratitude for the federal funding received by his city to manage the ongoing humanitarian crisis but emphasized the need for a long-term bipartisan solution.

El Paso is known for being a welcoming community, but Mayor Leeser stressed that the effort and resources expended on the crisis cannot continue indefinitely. He stated that a collaborative approach involving local, state, and federal partners, as well as other countries, is required to develop a more manageable, humane, and sustainable immigration system for the United States.

President Biden's plan is highly anticipated, and stakeholders at various levels of government are ready to work together to address the challenges posed by the broken immigration system. The current invitation to mayors indicates the administration's dedication to engaging with local leaders in finding effective solutions.

ABC News' Armando García contributed to this report.