On Wednesday, the House impeached Mr Trump on charges of abuse of power and obstruction of Congress.
But Democrats have yet to embark on the next stage, arguing the Republican-controlled Senate is refusing witnesses and will not hold a fair trial.
The Senate’s numbers mean Mr Trump is almost certain to be acquitted.
The impeachment process – only the fourth time it has happened in US history – has been a bitter partisan fight dividing Washington. The House impeachment vote earlier this week split almost totally along party lines.
The two charges passed on Wednesday follow accusations that Mr 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 Mr 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!”