On Wednesday, the House impeached Trump on charges of abuse of power and obstruction of Congress.
But Democrats have refused to start the proceedings, arguing the Republican-controlled Senate is refusing witnesses and will not hold a fair trial.
The Senate's numbers mean Trump is almost certain to be acquitted.
The impeachment process has been highly toxic and divided almost totally along party lines.
The two charges agreed on Wednesday follow accusations that Trump pressured Ukraine to dig up damaging information on Democratic rival Joe Biden and his son Hunter, and then refused to co-operate with a congressional inquiry into the matter.
What has Trump said?
In a series of tweets, the president accused the Democrats of not wanting to go to trial because their "case is so bad".
He tweeted: "So after the Democrats gave me no Due Process in the House, no lawyers, no witnesses, no nothing, they now want to tell the Senate how to run their trial. Actually, they have zero proof of anything, they will never even show up. They want out. I want an immediate trial!"
The president said the Democrats did not want Adam Schiff, who led the impeachment process, the Bidens and a CIA whistleblower who sparked the inquiry to testify.
Why is there deadlock over the start of the trial?
To get things rolling, the Democrat-controlled House must transmit the articles of impeachment to the Senate.
But House Speaker Nancy Pelosi is refusing to do so until the rules of the Senate trial are acceptable to the Democrats.
The Senate's Republican leader, Mitch McConnell, will determine the terms of the trial and the Democrats want him to provide details on which witnesses and what testimony will be allowed.
McConnell has refused to play ball. "We remain at an impasse," he said, after a brief meeting with the Democrats' Senate minority leader, Chuck Schumer.
McConnell has the numbers. There are 53 Republicans in the 100-seat Senate and impeachment would require a two-thirds majority in favour.
McConnell has called the impeachment process the "most rushed, least thorough and most unfair" in history, signalling the kind of bipartisan rancour expected when the trial starts.
The Democrats hope the delay will both increase public opinion for a fuller trial and deny Trump - only the third US president to be impeached - a swift acquittal.
The Democrats want at least four current and former White House aides with knowledge of the Ukraine affair to testify.
They say the trial has to be fair and that McConnell's comments show he has no plans to make it so.
What is the president accused of?
He is accused of having withheld $400m (£307m) of military aid to Ukraine already allocated by Congress, and a White House meeting for Ukraine's new president, until Ukraine looked into potentially damaging material on Joe and Hunter Biden.
Hunter worked for a Ukrainian company when Joe Biden was US vice-president.
The Democrats say this amounts to an abuse of presidential power, using the office for personal political gain and to the detriment of national security.
Trump is also accused of obstructing Congress by refusing to co-operate with the congressional inquiry.