Washington (CNN) Participants at a meeting between Republican senators and White House senior advisers Jared Kushner and Stephen Miller found the President's advisers' proposal to revamp the US legal immigration system to be lacking in substance, according to a senior GOP official familiar with the presentation.

"I think folks in the room were underwhelmed" with the "whole plan" presented by Kushner and Miller, the official told CNN.

A senior administration official pushed back on the criticism of the White House's presentation at the Tuesday GOP Senate briefing in which Kushner and Miller presented ideas under consideration by the White House to reform the nation's legal immigration system.

"The characterization that Kushner didn't know the subject matter is wrong on many accounts," the official said.

The meeting was the product of weeks of meetings with lawmakers and stakeholders in the immigration debate, and President Donald Trump is expected to unveil the details of the White House immigration plan as early as Thursday, a source familiar with the plan said.

The senior administration official said Kushner, along with Miller and White House economic adviser Kevin Hassett delivered separate portions of the briefing to represent different viewpoints on the subject. Kushner approached the immigration issue from the "left," Miller from the "right," while Hassett offered analysis that would appeal to the business community, the official added.

"We did it that way so the right wing agreed with the Miller piece," the official said. "The group did it together so there would be a whole of government approach."

Trump is expected to unveil the details of the White House immigration plan on Thursday, a source familiar with the plan said.

But one Republican aide told CNN after the meeting that the feeling in the caucus lunch was that immigration is a longshot at this point. Even doing something on the fringes would require tremendous time and energy that does not currently exist, said the aide who wasn't in the room but was briefed by the senator they work for who did attend.

The senator felt like everyone would like to see progress on this issue, the aide said, but that the fact there was no DACA or according to this source no E-Verify, makes it hard to imagine that this would end in any real, bipartisan deal.

"You have to be further along than what they presented," the aide who was briefed said.

Sen. Lindsey Graham, R-South Carolina, a close Trump ally who is presenting his own immigration plan, claimed "the White House's plan is not designed to become law."

Advocating for his own proposal, Graham said: "The White House plan is trying to unite the Republican Party around border security and merit-based immigration. I'm trying to get some relief to our Border Patrol agents. I'm trying to put a dent in the smuggling business and keep kids from going on a journey that's got to be hell."

Searching for a plan

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The President has pushed for a "merit-based" immigration system, which would prioritize high-skilled workers, and an end to chain migration.

He tasked Kushner and Nielsen to lead negotiations on border security funding. The first sessions of White House meetings with immigration groups and business groups began in January amid the government shutdown -- when Trump and lawmakers could not agree on new border wall funding. But Nielsen left her role as the head of Homeland Security in April, nudged out by border hardliners who felt she wasn't tough enough on stemming the flow of migrants to the border.

Some issues, such as Deferred Action for Childhood Arrivals, were not discussed at the meeting by design, the official added, to avoid discussion on topics that remain divisive.

The proposal also does not address family separations at the border, which have become a lightning rod issue for the administration.

However, the proposal also is likely to preserve family unification as a principle of the legal immigration system, meaning that immigrants will likely continue to be able to sponsor other immediate family members, helping them obtain legal status in the United States. This is, of course, the very system that Trump has derided as "chain migration."

"Unifying families is a basic tenant of immigration," the official said. "It's a whole component right now of a legal immigration system that's family based.

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Among the priorities of the proposal would be putting in place a merit "points" system for legal immigration. Reforming the legal immigration system will aim to keep the total number of legal immigrants entering the country the same, the official added.

It will also seek to secure ports of entry and increasing barriers, i.e. a wall, "where needed" across the border.

But other more conservative Republican lawmakers who received a briefing at the White House last week were more positive about the plan.

"If it adheres to the principles laid out in the meeting the plan would unite Republicans," one Senate aide said, acknowledging, however, that that's a much heavier lift in the Democratically-controlled House.

That aide added that the plan is a starting point to give Republicans "something to be for on immigration, not just be against."

One immigrant advocate who spoke with Kushner said he sounded open to the idea of providing aid to Central American countries where many of the migrants are fleeing their communities. But Kushner wanted to know whether the idea would be politically viable.

"Do you think that would poll well?" Kushner asked, according to the immigrant advocate.

Puzzling comments

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A representative of one organization who met with Kushner and former Homeland Security Secretary Kirstjen Nielsen about the White House proposal said both officials made puzzling comments when asked about the administration's plans for dealing with the surge of migrants at the border , a key sticking point in the immigration debate.

According to the representative, Nielsen had said the administration was considering the expansion of holding facilities for migrants to make room for activities aimed at helping migrants feel more at home while in detention.

The representative said Nielsen raised the idea of DHS officials arranging "Zumba" fitness classes for migrants while they're being held by immigration authorities.

"It was like 'let them eat cake,'" the representative said of the prospect of migrants taking "Zumba" exercise classes.

A DHS official told CNN some family centers run by US Immigration and Customs Enforcement already offer the fitness program to migrants in detention, so this would be an expansion of those opportunities.

During the meeting with Kushner, the representative said the President's son-in-law sounded optimistic about the prospect of Trump's border wall becoming a reality.

"We solved the butterfly thing," the representative quoted Kushner as saying.