Mick Mulvaney: No deal with Democrats on dreamers yet because Trump is still pushing the border wall

• White House budget director Mick Mulvaney says "no final deal" was made on the so-called dreamers with top Democrats.
• Mulvaney, who joined the president at a dinner with Democratic leaders Wednesday, also called a characterization that a deal is on the table is "fair."

White House budget director Mick Mulvaney told CNBC on Thursday "no final deal" was made on so-called dreamers with top Democrats at a White House dinner with President Donald Trump because the president is pushing his demand for a wall along the U.S.-Mexican border to crack down on illegal immigration.

Mulvaney, who was at the dinner Wednesday, also said "it's fair" to say that a deal on the Obama-era Deferred Action for Childhood Arrivals (DACA) program, which protected 800,000 young immigrants from deportation, is on the table. He added that while the discussion was very productive, the president has not given up on his immigration priorities including the wall that he promised during the campaign.

"The Democrats want to talk more and more about 'dreamers' and about permanent status. And the president said, 'Well, let's talk more about DACA.' And the president wanted to push the wall, very hard. And the Democrats said, 'Well, let's talk about border security,'" Mulvaney said on "Squawk Box."

"The reason that we can say here today that we don't have a deal is that the president is pushing the wall. Is it possible to get a deal without the wall now and perhaps getting the wall later ... that's the basis for conversation," added Mulvaney, formerly a GOP congressman from South Carolina and member of the hard-line conservative Freedom Caucus.

Mulvaney said it's not fair to say the president is giving up on something just because he's talking to Democrats. "We're just looking at this in bits and pieces," he said.

Following the dinner with Democrats Sen. Chuck Schumer and Rep. Nancy Pelosi, Trump said Thursday morning, "We're working on a plan for DACA," the Obama-era program that shielded immigrants brought illegally to the U.S. as children from deportation. Trump rescinded DACA on Sept. 5, but gave Congress six months to come up with a new plan.

On his way to Florida to survey the damage from Hurricane Irma, Trump said, "We're fairly close. But we need to get massive border security." He added: "The wall will be built."

Senate Minority Leader Schumer and House Democratic leader Pelosi said after the dinner that they reached a deal with the president on the Deferred Action for Childhood Arrivals program in exchange for border security measures but not the border wall.

Before speaking with reporters Thursday morning, Trump denied in a series of tweets that a deal with Schumer and Pelosi was reached.

No deal was made last night on DACA. Massive border security would have to be agreed to in exchange for consent. Would be subject to vote.

The WALL, which is already under construction in the form of new renovation of old and existing fences and walls, will continue to be built.

Does anybody really want to throw out good, educated and accomplished young people who have jobs, some serving in the military? Really!.....

...They have been in our country for many years through no fault of their own - brought in by parents at young age. Plus BIG border security

Mulvaney said tax reform was also discussed, adding the president's priorities for taxes are lowering the corporate rate, and cutting and simplifying taxes for working Americans. He said he got a sense at the dinner there could be a possibility of a deal on taxes, but added he's much more optimistic on an agreement on DACA and border security.