By Jeremy Herb, Manu Raju and Laura Jarrett, CNN

(CNN) -- Attorney General William Barr said Tuesday that special counsel Robert Mueller was given the opportunity to review the four-page summary Barr released last month detailing the conclusions of Mueller's investigation.

Mueller, who did not help Barr prepare the summary, declined the offer, Barr told a congressional subcommittee.

Barr also said the redactions process he's undertaking with Mueller to release the report was going "very well," and he expected the redacted report would be released within the next week. The redactions would be "color coded" into four categories, Barr said, and an explanation would be provided for the basis of each redaction.

"I do think it's important that the public have an opportunity to learn the results of the special counsel's work," Barr said.

Barr is testifying at a House appropriations subcommittee hearing to discuss the Justice Department budget and the department's priorities, but the Mueller report -- which Barr is in the process of redacting before releasing publicly -- looms over the proceedings.

"We cannot hold this hearing without mentioning the elephant in the room. Two and a half weeks ago the Mueller report was completed in extremely quick fashion. You turned a 300-plus page report into a 4-page letter that supposedly summarized the findings," said New York Rep. Jose Serrano, the chairman of the Commerce, Science and Justice Subcommittee holding Tuesday's hearing. "The American people want to see the report. I think it would strike a serious blow to our system, and yes, to our democracy, if that report is not fully seen. We're not here today to be in a confrontational situation with you. We want to help you do the job, and you need to help us do ours."

House Appropriations Chairwoman Nita Lowey told Barr that his handling of Mueller's report was "unacceptable" and the summary he released "raises more questions than it answers."

"I look forward to reviewing the Mueller report myself, and I know my constituents do as well. I understand that portions of it must be redacted as a matter of law, but my hope is that you will stop there and bring transparency to this process as soon as possible," Lowey said.

It's still unknown how much -- if anything -- Barr will say about the special counsel during his testimony Tuesday or when he returns for a Wednesday Senate appropriations hearing. Barr's nine-page written statement that was released Monday ahead of the hearing made zero mention of Mueller or the special counsel's report.

Barr's opening statement did not mention Mueller's report, instead outlining the Justice Department's budget priorities. But he knew questions about Mueller questions would be coming from Democrats.

"What's on everyone's mind right now is how much is he going to redact? ... He knows everyone is going to want to know what's behind the black ink," said Rep. Matt Cartwright, a Pennsylvania Democrat who will have a chance to question Barr at Tuesday's hearing.

Cartwright told CNN that the hearing would also address the committee's oversight responsibilities, but he was optimistic Barr would at least explain some of his thinking process when it comes to Mueller's report and findings.

Democratic aides say no topics are off limits for Tuesday's hearing, and they're likely to ask Barr about Mueller, as well as more traditional questions like the Justice Department budget, the Trump administration's lawsuit against the Affordable Care Act and gun violence. Committees at the center of the fight with Barr over the Mueller investigation and report have offered potential lines of questioning to Appropriations Committee Democrats, according to one aide.

Republican lawmakers, on the other hand, are not expected to ask questions about Mueller and will focus on budget issues in the hearing, according to a GOP aide. GOP Rep. Robert Aderholt of Alabama, the top Republican on the subcommittee, did not mention Mueller in his opening statement.

In addition to Mueller questions, Barr may also be pressed on the Justice Department's role in Trump's polarizing immigration policies, including its role in "zero tolerance" policies that led to the separation of immigrant families at the border, as well as the administration's request for additional immigration judges to ease a backlog of cases.

Even if Barr declines to speak about the special counsel probe at this week's budget hearings, he is coming back to Capitol Hill on May 1 and May 2 for hearings before the Senate and House Judiciary Committees specifically to answer questions about the Mueller investigation.

Barr told Congress earlier this month he expected to release a redacted version of Mueller's nearly 400-page report by "mid-April, if not sooner." He said four types of information would be redacted, including grand jury material, classified information, material tied to ongoing investigation, and information that could harm "peripheral third parties."

But Democrats said Barr's redactions plans are unacceptable, and they are demanding he provide Congress with the full, unredacted report and Mueller's underlying evidence. The House Judiciary Committee voted last week to authorize a subpoena for the full report and evidence.

House Judiciary Chairman Jerry Nadler has yet to issue the subpoena, but the New York Democrat says he will do so soon unless Barr provides the report, and he's willing to go to court to obtain it if the Justice Department will not provide the materials voluntarily.

This story has been updated with additional developments Tuesday.