Not backing away from the President's latest spat with a GOP Senator, the White House on Tuesday took square aim at the Congress, criticizing lawmakers for not passing legislation to overhaul the Obama health law, and urging fast action on a major tax reform bill, which Mr. Trump will press during a visit to Pennsylvania on Wednesday.

"The people of this country want tax cuts," the President said during a photo op with former Secretary of State Henry Kissinger, as Mr. Trump rejected the idea that his tussle with Corker would hurt his drive for tax legislation.

"People want to see massive tax cuts. I'm giving the largest tax cuts in the history of this country," the President added.

Corker was yet another Senate Republican to get a less than affectionate nickname from the President - back in the 2016 campaign, Mr. Trump routinely referred to Sen. Ted Cruz (R-TX) as "Lyin' Ted," and Sen. Marco Rubio (R-FL) as "Little Marco."

"I don’t think he's alienating anyone," Press Secretary Sarah Huckabee Sanders said of the President's jabs at Corker.

"I think that Congress has alienated themselves by not actually getting the job done that the people of this country elected them to do," Sanders told reporters.

"They all promised and campaigned on repealing and replacing Obamacare; they haven’t done that," Sanders said. "But time and time again, Congress has made promises and failed to deliver. If anyone is being alienated, it's people that are promising things and not delivering on them."

In Congress, Republicans keep talking optimistically about tax reform, but they have not yet revealed the exact details of their tax reform plan, while promising it will deliver major changes to the tax code.

Republicans say they will not unveil the fine print of the tax bill until the House and Senate have agreed on a budget outline for 2018.

The House approved its 'budget resolution' last week - the Senate is expected to vote on a different version next week. Negotiations may be needed to hash out the differences, moving the timeline for the release of the tax reform plan close to, if not into November.

Like the recent effort on a GOP bill to overhaul the Obama health law, this tax plan would only need a simple majority of votes in the Senate, as the expedited "budget reconciliation" process does not allow for a Senate filibuster.