Senate Republicans, with 22 seats up for reelection and their majority up for grabs in 2020, are in a bind over how to defend President Trump Donald John TrumpFederal prosecutor speaks out, says Barr 'has brought shame' on Justice Dept. Former Pence aide: White House staffers discussed Trump refusing to leave office Progressive group buys domain name of Trump's No. 1 Supreme Court pick MORE from the House impeachment push.

They realize their political fortunes and policy goals are tied to the president, but they also know their best shot at keeping control of the Senate hinges on senators preserving their independent brands.

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That means they have to be careful in defending conduct they see as improper; several Senate Republicans say it would be wrong to hold up military aid to Ukraine to gain political leverage, the issue at the heart of the impeachment inquiry.

Senate Republicans on Tuesday took a decidedly cautious approach as a White House witness offered new testimony in the House critical of Trump.

Senate Majority Leader Mitch McConnell Addison (Mitch) Mitchell McConnellDemocratic senator to party: 'A little message discipline wouldn't kill us' House to vote on resolution affirming peaceful transition of power Republican lawyers brush off Trump's election comments MORE (R-Ky.) claimed victory in Speaker Nancy Pelosi Nancy PelosiDemocratic senator to party: 'A little message discipline wouldn't kill us' Overnight Health Care: New wave of COVID-19 cases builds in US | Florida to lift all coronavirus restrictions on restaurants, bars | Trump stirs questions with 0 drug coupon plan Overnight Defense: Appeals court revives House lawsuit against military funding for border wall | Dems push for limits on transferring military gear to police | Lawmakers ask for IG probe into Pentagon's use of COVID-19 funds MORE’s (D-Calif.) decision to hold a House vote Thursday on impeachment procedures but stopped short of outlining his next moves.

“Obviously they’ve responded to the pressure that we put on them to try to handle this in a more transparent way that meets basic standards of due process that every American would be entitled to,” he said.

“We’ll have to take a look at what the House produces later today and see if it passes the smell test of providing the kind of due process protections that the president and his team are certainly entitled to, just like President Nixon was and President Clinton was,” he said.

Trump and Republican leaders in the House have telegraphed a new strategy intended to move away from attacks against Democrats on process, and toward a more robust defense of the president on the merits of impeachment.

“I’d rather go into the details of the case rather than the process,” Trump said Monday before a trip to Chicago.

“Process is wonderful,” he added. “But I think you ought to look at the case.”

But McConnell, a master of procedural fights, didn’t endorse the shift in his comments.

Instead, he noted that the resolution he and 49 other Republican senators sponsored condemning the House impeachment process “didn’t deal with the merits” of the charges against Trump.

And McConnell seemed reluctant to give up the strategy of attacking Democrats on procedural grounds.

Several factors help explain McConnell’s approach, while illustrating the tight spot he is navigating.

The GOP conference he leads is divided over the substance of the charges against Trump.

Three Senate Republicans, Susan Collins Susan Margaret CollinsDemocratic senator to party: 'A little message discipline wouldn't kill us' Poll: 57 percent of Americans think next president, Senate should fill Ginsburg vacancy On The Trail: Making sense of this week's polling tsunami MORE (Maine), Lisa Murkowski Lisa Ann MurkowskiDemocratic senator to party: 'A little message discipline wouldn't kill us' Overnight Energy: Trump officials finalize plan to open up protected areas of Tongass to logging | Feds say offshore testing for oil can proceed despite drilling moratorium | Dems question EPA's postponement of inequality training Poll: 57 percent of Americans think next president, Senate should fill Ginsburg vacancy MORE (Alaska) and Mitt Romney Willard (Mitt) Mitt RomneyCrenshaw looms large as Democrats look to flip Texas House seat The Hill's Morning Report - Sponsored by Facebook - Republicans lawmakers rebuke Trump on election Trump dumbfounds GOP with latest unforced error MORE (Utah), did not support the resolution condemning the House inquiry.

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A senior Republican senator predicted last week the resolution would never get a vote on the Senate floor because it would put vulnerable incumbents and GOP senators who have expressed concerns over Trump’s conduct in a tough position.

McConnell on Tuesday said, “We haven’t made a decision on that yet.”

Polls of independent voters are finding more support for impeachment, a trend undoubtedly troubling to McConnell — who is focused on ensuring the GOP keeps its majority next year.

A Reuters-Ipsos poll last week found support for impeaching Trump surging among independents to 45 percent, with just 32 percent opposing it. A survey from Quinnipiac University Poll last week showed a 16-point swing in favor of the House impeachment investigation among self-described independents.

Republican senators are also uncomfortable defending Trump on the underlying charge, that Trump held up military aid to Ukraine to pressure its leaders to investigate former Vice President Joe Biden Joe BidenFormer Pence aide: White House staffers discussed Trump refusing to leave office Progressive group buys domain name of Trump's No. 1 Supreme Court pick Bloomberg rolls out M ad buy to boost Biden in Florida MORE.

The attacks by Trump allies on key House witnesses who have impressive résumés and no obvious signs of partisan affiliation make GOP senators uncomfortable.

McConnell on Tuesday said he did not want to question the patriotism of Lt. Col. Alexander Vindman, who testified before the House on Tuesday about Trump’s July 25 phone call with Ukrainian President Volodymyr Zelensky, and other witnesses.

“I’m not going to question the patriotism of any of the people who are coming forward,” he told reporters.

Vindman testified that he thought Trump’s pressure on Zelensky was improper and worried about the implications for U.S. support for Ukraine.

Finally, if the House does go forward with impeachment, the Senate will be the jury. Several GOP senators have indicated they see their role in the process as serious, and as a result do not want to go too far with their public statements.

Senate Republicans largely took a step back from the impeachment debate Tuesday. A GOP senator who attended the weekly lunch meeting said impeachment did not take up much discussion time.

Senate Republican Policy Committee Chairman Roy Blunt Roy Dean BluntSCOTUS confirmation in the last month of a close election? Ugly Senate to push funding bill vote up against shutdown deadline Social media platforms put muscle into National Voter Registration Day MORE (Mo.), who hosted the lunch, said “on impeachment, it’s just totally up to the House.”

“We’ll deal with it if the House sends it over, and we’ll see what the calendar in front of us looks like, and whether holidays or the Democratic caucuses or anything else has any impact on how the Senate wants to deal with it timewise,” he said. “Otherwise we’ll deal with it exactly as the Senate rules require.”

McConnell on Tuesday largely dodged a question about whether he considers the House impeachment process as “illegitimate.”

“Look, impeachment as a practical matter is whatever a majority of the House decides it is at any given moment,” he said.

“It appears that the Democrats, emboldened by the new majority they’ve had this year, have been on this path for three years,” he added, sticking to his general defense of Trump by arguing the Democrats are driven primarily by politics and a desire to overturn the results of the 2016 election.

Jordain Carney contributed.