In this shift in strategy to defend Mr Trump, these Republicans are insisting that the president's action was not illegal and does not rise to the level of an impeachable offence as the Democratic-led House moves forward with the open phase of its probe.
But the shift among Senate Republicans could complicate the message coming from Mr Trump as he furiously fights the claim that he had withheld US aid from Ukraine to pressure it to dig up dirt on a political rival, even as an increasing number of Republicans wonder how long they can continue to argue that no quid pro quo was at play in the matter.
The pivot was the main topic during a private senate GOP lunch on Wednesday, according to multiple people familiar with the session who spoke on the condition of anonymity to describe the meeting. Senator John Neely Kennedy argued that there may have been a quid pro quo but said that the US government often attaches conditions to foreign aid and that nothing was amiss in Mr Trump's doing so in the case of aid to Ukraine, these individuals said.
Inside the lunch, Senator Ted Cruz who ran against Trump in 2016, said a quid pro quo is not illegal unless there is "corrupt intent" and echoed Mr Kennedy's argument that such conditions are a tool of foreign policy.
Trump impeachment: Who's who in the Ukraine scandal
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"To me, this entire issue is gonna come down to, why did the president ask for an investigation," Mr Kennedy, who worked as a lawyer, said in an interview. "To me, it all turns on intent, motive. ... Did the president have a culpable state of mind? . . . Based on the evidence that I see, that I've been allowed to see, the president does not have a culpable state of mind."
The discussion underscores the dilemma for congressional Republicans as a cadre of current and former Trump administration officials paint a consistent picture of a president wiling to use foreign policy to undercut a potential domestic political adversary. On Thursday, Trump appointee and longtime Republican aide-turned-National Security Council adviser Tim Morrison became the latest official to testify that nearly $400m (£309m) of congressionally appropriated military aid for Ukraine was frozen to increase pressure on President Volodymyr Zelensky to investigate Mr Biden, a 2020 presidential contender.
And with the House Democrats voting Thursday to open the closed-door impeachment investigation, undermining the GOP's complaints about a secretive process, Republicans are frantically seeking a new strategy and talking points to defend the president.
Meanwhile, the president has frustrated senate Republicans by seeming to change his messaging strategy every day rather than present a coherent defence of his actions, said multiple senate GOP officials who spoke on the condition of anonymity to comment frankly.
On Thursday, Mr Trump told the Washington Examiner that he wanted to do a series of "fireside chats" - as President Franklin D Roosevelt famously did during the Great Depression and the early part of World War II - to defend himself. He said he perhaps would read aloud the transcript of the 25 July telephone call in which he asked Mr Zelensky to do him a "favour."
The willingness of some senate Republicans to acknowledge a quid pro quo while dismissing the offence comes just two weeks after acting White House chief of staff Mick Mulvaney had to walk back a similar assertion. Mr Mulvaney argued in a long-winded news conference on 17 October that quid pro quos are a common feature of foreign policy and that the media should "get over it."
Congressional Republicans balked, forcing Mr Mulvaney to retreat.
Indeed, a strategy that includes acknowledging a particular kind of reciprocity with a foreign government would almost certainly unnerve moderate Republicans such as Senator Susan Collins of Maine, who faces re-election next year in a Democratic-leaning state and has said it was "completely inappropriate" for Mr Trump to invite China to investigate Joe Biden, which the president did after the Ukraine controversy began. Ms Collins has repeatedly declined to comment on Ukraine, arguing that she probably would be a juror in a Senate trial of the president.
One senior Republican aide cautioned that acknowledging a quid pro quo is unlikely as a strategy for the senate GOP, even if some conservatives like the idea.
Such a step would also undercut Mr Trump's central talking point on impeachment - and would clash with house Republicans' strategy. Mr Trump's Capitol Hill allies and Republican leaders, including house minority leader Kevin McCarthy of California and house minority whip Steve Scalise of Louisiana, are sticking with Mr Trump's line that there was no proposed trade-off with Ukraine.
"You can't have been in [the impeachment depositions] with 10 different witnesses and come out with any credible belief that there was a quid pro quo for aid. . . . It's just not accurate," said top Trump ally Republican Mark Meadows. "I've heard people say, 'Well even if he did it, it's fine.' The problem with that is: I know that he didn't do it."
In the Senate, however, some Republicans aren't as confident and have expressed concerns about the endless drip of embarrassing headlines from daily witness testimony that the US aid and a White House visit for Mr Zelensky hinged on the Biden probe.
The senate lunch, according to those in attendance, also focused on how best to rally to Mr Trump's defence if he is impeached. Under the constitutional process, the Senate would hold a trial, with conviction requiring the votes of two-thirds of the senators present.
While some senate Republicans have argued for a quick trial, most other senators believe that moving quickly could backfire. Senate Republicans, especially those up for re-election next year in Democratic-leaning or swing states, could face criticism that they did not take the charges seriously.
During last year's contentious battle to confirm now-Supreme Court Justice Brett Kavanaugh, Republicans thought that the more thorough process, including a new FBI investigation in the final days of the confirmation fight, arguably helped to win more support, including that of Collins and then-Senator Jeff Flake.
Some Republicans suggested a longer trial could help the president by giving the GOP the opportunity to try to poke holes in the Democrats' case. Among those who made a case for thorough proceedings was Senator Mike Lee who responded to a remark from Senator Kevin Cramer that while, in theory, White House attorneys could move every day to try to dismiss the case, the administration would be advised not to do that.
Mr Cramer appeared to agree with that argument.
"This may be his only opportunity to change what the public sees and hears if they're gonna continue with their very one-sided process over in the House," Mr Cramer said, later adding: "In my view, [it is] in the president's best interest to have the whole thing played out. I don't mean five weeks, but at least the case so at least the public gets to hear his case."
Mr Cramer also subscribed to the idea that "there's lots of quid pro quos" in US foreign policy, pointing to stipulations on assistance for Venezuela and other nations.
"We've done quid pro quos a lot of times," he said. "... The question isn't whether it was quid pro quo; the question is: Was it corruption?"
Mr Kennedy argued that there are two views on Mr Trump's actions: that he pushed for an investigation of a political rival; or that he pushed for an investigation of corruption in a country that has a history of mis-steps - and the request just happened to include the Bidens. Mr Kennedy, an ally of Mr Trump's, said he expects the president's lawyers to argue the latter during a senate trial.
Mr Trump asked Mr Zelensky in the call to look into the Bidens, referring to allegations that Joe Biden pressured Ukrainian officials to fire a prosecutor who was probing the company where Hunter Biden served as a director. Former Ukrainian and US officials say the prosecutor's investigation into the company was dormant.
"He honestly believes that there may have been corruption in Ukraine, and before he turns over $400m of American taxpayer money, he's entitled to ask," Mr Kennedy said, later adding, "The issue to be litigated . . . is going to be: Did the president have a good-faith reason to believe that Hunter Biden may have been involved in corruption? And if I'm correct in my analysis, then there will be a lot of time spent on what Mr Biden did for the money."
Senator Ron Johnson who has been heavily involved in the Ukraine saga and is chairman of the Foreign Relations subcommittee on Europe and regional security cooperation, argued that the United States often puts conditions on foreign aid.
Mr Johnson also said that Mr Trump would not allow him to promise Ukrainian officials military aid earlier this year because of other reasons, including concerns about corruption and the desire for European nations to do more to help Ukraine.
"My point is those are legitimate reservations," he said. "There's nothing wrong with that. . . . That's not impeachable."
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