Biden to sign executive order on immigration as early as this week: Sources

President Joe Biden is expected to sign an executive order on immigration as early as Tuesday, according to sources familiar with the decision.

The long-awaited executive order would limit the number of migrants that would be allowed to claim asylum at the southern U.S. border. It would immediately send them back to their country of origin to wait until the daily average goes down and, once it goes down, they would be able to claim asylum, according to a source familiar.

The daily number of encounters has to hit 2,500 for the provision to kick in, according to sources briefed on the situation, meaning that one day of more than 2,500 migrants would trigger the provision of limiting asylum at the border.

In recent days, members of Congress have been briefed on the executive action, according to sources familiar with the briefings.

Republican lawmakers on Tuesday, ahead of the expected announcement, were highly critical of the decision -- despite their repeated calls for Biden to use his authority to address the surge of migrants after the GOP tanked a more comprehensive bipartisan border bill at former President Donald Trump's direction.

Speaker Mike Johnson pointed to Biden's own statement back in January that he'd "done all I can do" with regard to presidential power on the issue.

“Now suddenly he wants to issue some weak executive order," Johnson said at his weekly press conference with other GOP leaders. "It’s window dressing. Everyone knows it.”

Across the Capitol, a group of Republican senators gathered for their own news conference to similarly blast the executive action as "too little too late."

"It is a shell game. They are not serious about it," said Texas Sen. John Cornyn. "This is a conversion baked on the proximity of the next election and sinking poll numbers. And we think it deserves to be called out for what it is."

Johnson was pressed by a reporter about their opposition to Biden's decision, asking the speaker why no limit would be better than the limit Biden is seeking to implement.

“Limits are fine but it's not going to solve the problem,” Johnson claimed.

Any executive order, administration officials caution, would be challenged in court.

"I anticipate that if the president would take executive action, and whatever that executive action would entail, it will be challenged in the court," Homeland Security Secretary Alejandro Mayorkas told reporters last month at Department of Homeland Security headquarters.

Mayorkas and other members of the administration have urged Congress to pass the bipartisan border bill that was negotiated and proposed earlier this year.

A spokesperson for Brownsville, Texas, Mayor John Cowen confirmed to ABC News that the White House invited him to a meeting at the White House on Tuesday for an immigration-related announcement, and he will be attending.

El Paso Mayor Oscar Leeser also confirmed he is attending. He told ABC News in a statement: "El Paso is a welcoming community, and that makes me very proud, but no community can continue the effort and resources we've expended on this humanitarian crisis endlessly. We are appreciative of the funding we have received from the federal government so that our efforts don't fall on the backs of El Paso taxpayers, but our immigration system is broken, and it is critical that Congress work on a bipartisan long-term plan to work with other countries in order to create a more manageable, humane and sustainable immigration system for our country.

"I look forward to hearing more about the president's plan on Tuesday, and we stand ready to work with our partners at the local, state and federal level on this effort," he added.

ABC News' Armando García and Alexandra Hutzler contributed to this report.