Cummings of Maryland, the top Democrat on the House Oversight and Government Reform Committee, also said based on an unreleased 6-hour congressional interview last week with an anonymous 21-year IRS manager who called himself a "conservative Republican," "the case is solved."
The veteran manager said he started the targeting of conservative groups and, contrary to GOP allegations, the White House wasn't involved and it wasn't politically motivated, Cummings told CNN's "State of the Union" Sunday.
"I think this interview and these statements go a long way to ... showing that the White House was not involved in this," he said.
"If it were me, I would wrap this case up and move on, to be frank with you."
An IRS inspector general's report last month found the IRS used inappropriate criteria, such as the words "Tea Party" and "patriot," to target certain groups seeking tax-exempt status for extra scrutiny. Those criteria were in place for more than 18 months, resulting in substantial delays in processing targeted applications, the report said.
Cummings told CNN he asked Issa, a California Republican and the oversight committee's chairman, to release transcripts of the IRS manager's interview.
"I want those transcripts to be released," Cummings said. "But he's the chairman of the committee. We're not in power. Now, if he does not release them, I will. Period."
When asked for a time frame, he said he would "release them by the end of the week" if Issa doesn't.
Cummings, who has generally had a pleasant public relationship with Issa, also criticized Issa for continuing to allege the targeting was politically motivated, even after the IRS manager's testimony.
"Chairman Issa has a tendency to make strong allegations and then go chasing the facts, and usually never finding them," Cummings told the CBS News program "Face the Nation."
Cummings began the day sending Issa a letter he also released publicly alleging Issa's "actions over the past three years do not reflect a responsible, bipartisan approach to investigations, and the committee's credibility has been damaged as a result."
The letter can be seen at tinyurl.com/Cummings-Issa-Letter.
Issa said in a statement Cummings' TV comments about the IRS manager's interview didn't negate other committee interviews with two Cincinnati IRS employees who said a career IRS lawyer in Washington helped direct the early stages of the targeting between March and October of 2010.
Top IRS officials earlier blamed the targeting on lower-level workers in Ohio.
"The only thing Ranking Member Cummings left clear in his comments today is that if it were up to him the investigation would be closed. Fortunately, the decision to close the investigation is not his to make," Issa said.
"His extreme and reckless assertions are a signal that his true motivation is stopping needed congressional oversight and he has no genuine interest in working, on a bipartisan basis, to expose the full truth."
Issa's statement did not say if he intended to release the interview transcripts.