President Trump said this weekend that he offered a “wonderful deal” to protect illegal immigrant Dreamers and that resistance among Democrats suggests the party would rather have a political issue than a solution.

Democrats have largely rejected the outlines of Mr. Trump’s proposal, saying that while the legalization of 1.8 million Dreamers was an important step, they would not agree to his plans for a border wall or to limiting the long chain of family migration.

The president is poised to deliver his plan to Capitol Hill this week and make his case directly to the public in his State of the Union address.

Mr. Trump took to Twitter late Saturday to say part of the reason he offered to protect 1.8 million Dreamers — nearly three times the number protected from deportation under the Obama-era Deferred Action for Childhood Arrivals amnesty — was to try to make Democrats feel the heat.

“I have offered DACA a wonderful deal, including a doubling in the number of recipients & a twelve year pathway to citizenship, for two reasons: (1) Because the Republicans want to fix a long time terrible problem. (2) To show that Democrats do not want to solve DACA, only use it!” Mr. Trump said.

“Democrats are not interested in Border Safety & Security or in the funding and rebuilding of our Military. They are only interested in Obstruction!” he added.

His plan would couple the amnesty for Dreamers with a $25 billion request for wall funding, limiting family sponsorship of migration to spouses and minor children, eliminating the Diversity Visa Lottery that gives away immigration passes based on chance, and a number of policy changes to stop abuse of the asylum system and speed deportations of illegal immigrants.

Republicans said that plan could be the basis of talks but they are eager to entice Democrats to deal.

“I think whatever we do is not going to pass with one party or the other,” House Majority Leader Kevin McCarthy, California Republican, said on NBC’s “Meet the Press.” “If you want to solve this and not be at this in another five or 10 years, you want to do it correctly and I think that bill has a lot of merit to solving the problem.”

Sen. Lindsey Graham, a South Carolina Republican who proposed a plan this month that would have protected 3 million illegal immigrants, said Mr. Trump’s proposal for 1.8 million Dreamers is a “credible offer.”

But he said the president was asking for too much on the chain migration side.

“We’re going to get there. We’re going to give a pathway to citizenship for 1.8 million Dreamers who have been well-vetted. We’re going to do something substantial on the border. We’re not going to cut legal immigration in half,” he said on ABC’s “This Week” program. “We’ll deal with chain migration in two phases, beginning in phase one.”

He also urged Democrats to “calm down” after several days of over-the-top vitriol that included accusations of racism and xenophobia lobbed at the president.

House Minority Leader Nancy Pelosi, California Democrat, has said the proposal is part of Mr. Trump’s plan to “make America white again,” and Senate Minority Leader Charles E. Schumer, New York Democrat, said Mr. Trump was using the Dreamers as a tool to fulfill a wish list for immigration hard-liners.

Sen. Bernard Sanders, Vermont independent, said Sunday that he has serious concerns about some parts of the president’s proposal and that the focus should be on ensuring legal status and a path to citizenship for Dreamers.

“Where the bad part comes is the idea of a wall, which I thought was a great idea in the 15th century when China built the Great Wall. Not so smart today when we have technology that is much more effective,” Mr. Sanders said on CBS’s “Face the Nation.”

“I think you’re going to see a lot of debate about how much money we should spend on border security,” he said.

Rep. Will Hurd, Texas Republican, and Rep. Pete Aguilar, California Democrat, said Mr. Trump was too ambitious and that the better solution is a narrow bill that protects Dreamers but doesn’t do nearly as much on immigration policy or border security.

They have offered a bill that would couple an amnesty with calls for more studies of border security and a promise for action three years from now.

“This is the only bill that would have 218 votes on the House floor. We feel very confident about that,” Mr. Hurd said on CBS.

In the Senate, meanwhile, Sen. Joe Manchin III, West Virginia Democrat, said hopes for a solution might lie in the two-dozen-member Common Sense Coalition of moderate senators who recently worked to end the three-day partial shutdown of the federal government.

“The president has laid out a template. We’re going to look at that template, and we’re going to work with it,” Mr. Manchin said on CNN’s “State of the Union.”

He also said comments like Mrs. Pelosi’s aren’t helpful.

“We don’t need that type of rhetoric on either side — from Nancy, Paul Ryan or anybody else,” Mr. Manchin said.

“I’m condemning all this crazy rhetoric that goes on,” he said. “We have to work together.”