Updated

President Trump met Tuesday with Senate Republicans in hopes of forging the kind of party unity needed to pass tax reform and other legislative goals before next year’s congressional elections, even as he and key lawmakers clashed on the sidelines.

In the lead-up to Tuesday's meeting, Trump and Sen. Bob Corker, R-Tenn., reignited a simmering feud with a round of barbed insults.

Corker and Sen. John McCain, R-Ariz. -- another occasional Trump adversary -- both emerged from the meeting Tuesday afternoon without answering questions from reporters.

However, by Tuesday night Trump sounded optimistic. "So nice being with Republican Senators today. Multiple standing ovations! Most are great people who want big Tax Cuts and success for U.S.," he tweeted.

Louisiana GOP Sen. John Kennedy told reporters that Trump and Corker did not speak directly during the meeting, which he described as "very cordial." Kennedy said the meeting ended with standing ovations.

The GOP senator said tax reform, as expected, was the main topic, even as lawmakers discussed infrastructure and health care with the president as well.

However, he said Trump set no deadline for Congress to send a tax bill to his desk.

"We had a very, very good meeting," Kennedy said, urging Trump to travel across the country to sell a tax plan. Asked repeatedly about the Corker feud, Kennedy said there were no conversations about Trump's tweets.

With the tax plan, Trump is still trying for a major legislative win after Republicans failed this summer to pass legislation to dismantle ObamaCare. The Senate passed a related budget plan last week -- but specifics on the tax legislation are still being worked out, including over tax rates and provisions for popular benefits like pretax 401(k) contributions.

But hopes of unity faced another setback Tuesday morning when Corker reignited his feud with Trump.

“I would just like him to leave it to the professionals for a while and see if we can do something that’s constructive,” Corker said on ABC’s “Good Morning America” about overhauling the federal tax code for the first time in roughly three decades.

Trump responded with a series of vitriolic tweets including: “Isn’t it sad that lightweight Senator Bob Corker, who couldn’t get re-elected in the Great State of Tennessee, will now fight Tax Cuts plus!”

The president appeared last week to settle his differences with Senate Majority Leader Mitch McConnell, R-Ky., emerging from a White House meeting to say his relationship with the Senate Republican leader is “better than ever.”

However, McCain and other GOP senators still appear at odds with Trump, who essentially needs consensus from the 52-member Senate Republican conference to get a simple 51-vote majority to pass tax reform and most other legislative proposals.

The Senate narrowly passed its blueprint plan last week, 51-49. And McCain cast the deciding no vote in August that nixed the chamber’s ObamaCare repeal-and-replace bill.

McCain on Sunday took a swipe at Trump when he said on C-SPAN that the U.S. military has “drafted the lowest income level of America and the highest income level found a doctor that would say they had a bone spur."

Trump had five draft deferments, including one granted after he submitted a physician's letter stating he suffered from such spurs in his feet.

House Speaker Paul Ryan, R-Wis., said Tuesday he expects his chamber to have its final tax bill ready by the Thanksgiving holiday but that the Senate is going to be “a little slower,” which would mean the final package would likely be ready by year’s end.

“At the end of the day … Bob’s going to vote for tax reform because he knows it’s in the best interest of Americans. So put this Twitter dispute aside,” Ryan also said about the Corker-Trump feud.

Among the issues still being debated in the effort to overhaul the tax code is the future of employee retirement plans known as the 401(k)s.

Congress is looking for ways to raise revenue as part of an overhaul plan. And one of the methods reportedly under consideration is to curtail how much pretax money workers can contribute to their 401(k) and similar accounts. Such a move would strike at a way that tens of millions of Americans use to save for retirement.

The suggestion has already run into some resistance, even if it isn't an official policy proposal.

Trump said Monday in a tweet: "There will be NO change to your 401(k). This has always been a great and popular middle class tax break that works, and it stays!"

Another key issue is the new corporate tax rate, which is now at 35 percent and that the administration would like to have at 20 percent.

In addition, congressional Republicans are debating whether the new tax code will have a fourth tax bracket, for the country’s highest wage-earners.

Fox News' Joseph Weber and The Associated Press contributed to this report.