Has the White House expressed willingness to revisit the January 19, 2018 DACA deal?

Question

As described in this question and answer, Senate Minority Leader Chuck Schumer claims that on Friday (January 19, 2018) he offered a deal to President Trump that would have averted the government shutdown, included DACA, and included $20 billion of funding for the Wall, but President Trump turned it down. But as you can see here, the White House is disputing this, saying that Schumer only offered $1.6 billion for Wall rather than $20 billion, but Schumer's spokesperson said the White House wasn't telling the truth about this.

This seems like a fairly easy problem to solve: just pass the Friday deal with $20 billion for the wall. My question is, has the White House expressed willingness to do this?

By the way, history seems to be repeating itself. In 2011, President Obama and Speaker of the House John Boehner were close to agreeing on a "grand bargain" that would have included $2.4 trillion in spending cuts in exchange for $800 billion in revenue increases. Then President Obama asked whether Boehner was willing to do $1.2 trillion in revenue instead (after Senate Republicans had expressed willingness to do so). Boehner refused and pulled out of negotiations. When asked by the press afterwards whether he'd be willing to go back to the $800 billion deal, Boehner said "you can't put Humpty Dumpty back together again". I suspect something similar may be happening here.

Answer 1

Yes, shortly after you asked your question, the White House came with a counter offer. From the Associated Press, dated 26 of January 2018 (emphasis mine):

WASHINGTON (AP) — The White House unveiled a proposal Thursday that provides a pathway to citizenship for 1.8 million young immigrants living in the country illegally, in exchange for new restrictions on legal immigration and $25 billion in border security.

Specifically, from the same source:

The plan would provide a pathway to citizenship for the roughly 690,000 younger immigrants protected from deportation by the Obama-era Deferred Action for Childhood Arrivals program — as well as hundreds of thousands of others who independent estimates say qualify for the program, but never applied.

[...]

The plan would not allow parents of those immigrants to seek lawful status, the officials said. In exchange, Trump’s plan would dramatically overhaul the legal immigration system. Immigrants would only be allowed to sponsor their spouses and underage children to join them in the U.S., and not their parents, adult children or siblings.

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In the end, the legislation that came in the aftermath of the shutdown was the Consolidated Appropriations Act, 2018. Wikipedia references some things that are not covered by it but were speculated (I will cite a few that we saw in this answer previously or that are related otherwise):

The bill includes $1.6 billion for Mexico–United States border wall, less than the $25 billion sought by the Trump administration. The funding is largely restricted to upgrading existing fencing and planning and design.

[...]

The bill did not fund the hiring of hundreds of new Border Patrol agents and immigration-enforcement agents.

[...]

The bill did not have penalties to cities that do not enforce federal immigration laws.