A Senate Republican bill endorsed by President Trump that would drastically reduce legal immigration would have a tough time getting full GOP support, Republicans acknowledged on Wednesday.

"If we get on immigration, there will be a lot of different perspectives on that," Senate Republican Conference Committee John Thune, R-S.D., told the Washington Examiner. "We have workforce needs, and some of those are filled by the immigrant labor supply."

A bill introduced by Sens. Tom Cotton of Arkansas and David Perdue of Georgia would establish a skills-based system for admitting immigrants, slash the number of refugees admitted each year, and cut legal immigration by half.

Trump enthusiastically promoted the bill at the White House Wednesday. Trump said he campaigned on a merit-based immigration system, which he argued would protect employment prospects for Americans and lower spending on immigrants who come here without the means to survive. The Cotton-Perdue bill, Trump said, would "reduce poverty, increase wages, and save taxpayers billions and billions of dollars."

Back on Capitol Hill, senators were more cautious, particularly those from states that have become reliant on an immigrant population to work in agriculture.

"In Wisconsin, not one dairy plant can hire enough people to work," Sen. Ron Johnson, R-Wis., told the Washington Examiner. "We do have to be concerned about people with that skill level, who work in manufacturing and agriculture."

But Johnson and other lawmakers pointed to aspects of the bill they could support.

"We have to have skills, merit-based immigration system," Johnson said. But he added, "I'm a little concerned with the numbers they are talking about."

Johnson said he is compiling a history of people who have been nationalized and who have come to the United States legally.

"What is the composition of that, what is the requirement?" Johnson said. "I don't want to restrict legal immigration, but I don't want to bring it to the point where we are restricting American wages."

Johnson is advocating for a temporary guest worker program managed by the states, which would give them the power to determine workforce needs. The provision could melded with the Cotton-Perdue proposal, Johnson said.

"Maybe there is a combination of something that would work well on that," Johnson said.

Senate Majority Whip John Cornyn, R-Texas, didn't promise the bill would make it onto the Senate floor schedule, but said he backs it and expected it to be considered in committee.

Cornyn said past immigration debate has included studying merit-based systems like the one used by Canada. The Senate's 2013 bipartisan immigration reform plan included a merit-based system for visas.

"It's an unresolved mess for the country," Cornyn said. "And I think it is worth discussion. I think it's a good argument. And what that level of immigration should be, that is a discussion we ought to have."

Republicans, however, may be nowhere near ready to tackle immigration. A plan fell through to pass a bill to repeal and replace Obamacare by the August recess, and now the effort has stalled.

Republicans have moved on to tax reform, a massive endeavor that could prove even harder to accomplish. On Wednesday, nobody seemed eager to add immigration reform to the list.

"Healthcare and taxes is tough enough," said Sen. Johnny Isakson, R-Ga.