The Democrat-backed inquiry is focusing on a July 25 telephone call in which Mr Trump allegedly urged Ukrainian President Volodymyr Zelenskiy to investigate political rival Joe Biden. Mr Biden, who served as vice president under Barack Obama, is a leading contender for the Democratic presidential nomination to face Mr Trump in 2020 and US law prohibits candidates from accepting foreign help in an election.
But the Republicans have introduced a US Senate resolution branding the impeachment process unfair.
And 44 of the 53 Republicans in the 100-seat Senate have signed on to the resolution which urges the Democratic-controlled House of Representatives to hold a formal vote to initiate the impeachment inquiry and give Mr Trump the ability to “confront his accusers” and call his own witnesses.
Lead sponsor Senator Lindsey Graham said: “I’m not here to tell you that Donald Trump’s done nothing wrong.
“I’m not here to tell you anything other than that the way they’re going about it is really dangerous for the country and we need to change course while we can in the House.”
Mr Graham described efforts to indict Mr Trump “out of bounds” and a danger to the US presidency.
He said: “What you’re doing today, in my view, is unfair to the president, is dangerous to the presidency, and I think 44 Republican senators and growing is a strong signal to our House colleagues that you’re off script here.”
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Mr Graham, who said he agreed with the president’s description of the impeachment probe as a “lynching,” also branded the House process “illegitimate.”
If the House passes articles of impeachment – formal charges – the Republican-controlled Senate would then hold a trial on whether to remove Mr Trump from office. A two-thirds majority would be required to remove the president.
Only three US presidents before Mr Trump have faced impeachment inquiries and none were removed from office via the process.
Republicans have complained about Democrats hearing from witnesses in closed sessions and not allowing lawmakers, who are not members of the three committees leading the inquiry, from being present for depositions.
But top Senate Democrat Chuck Schumer accused Mr Trump and his Republican congressional allies of hypocrisy for demanding that the proceedings be held in public even as the administration withholds subpoenaed documents and tries to block testimony to House investigators.
Mr Schumer said: “The hypocrisy, the self-serving nature of the president’s statements and his Republican allies here in the House and Senate is glaring.”
Democrats have said there will be public hearings in the coming weeks.
They hope to complete the inquiry by the end of the year and are coalescing around two articles of impeachment: abuse of power and obstruction.
Mr Trump withheld £300 million in security aid to Ukraine passed by Congress to help counter Russia-backed separatists in eastern Ukraine.
During the call with Mr Trump, the Ukrainian president agreed to investigations of Mr Biden and his son Hunter Biden’s tenure on the board of Burisma, a Ukrainian energy company.
The aid money was later released.