US Democrats have ruled out a "witness swap" with Republicans in President Donald Trump's impeachment trial.
Lawmakers who are seeking to remove the president from office hope to hear testimony from his former National Security Adviser John Bolton.
But Democrats refused any deal to allow the son of former US Vice-President Joe Biden to be called as a witness.
Mr Trump is accused of abuse of power and obstruction of Congress. He strongly denies any wrongdoing.
House Democrats have up to three days to make their case as they present their arguments in the impeachment trial in the Senate. Mr Trump's defence team will have three days after that for a rebuttal.
Democrats accuse the president of using US military aid as a bargaining chip in an attempt to prod Ukraine into announcing an investigation to discredit his would-be Democratic White House challenger, Mr Biden.
Mr Trump has been touting corruption claims against Mr Biden, whose son Hunter held a lucrative board position with a Ukrainian gas firm while his father was US vice-president and in charge of American-Ukrainian relations.
Attending the World Economic Forum in Davos, Switzerland, on Wednesday, Mr Trump jokingly warned he might confront Democrats by coming to "sit right in the front row and stare at their corrupt faces".
The impeachment trial could end next week, but Mr Trump's fellow Republicans control the chamber and are unlikely to oust him.
Goading by Trump?
There was one thing in particular that President Trump said which was kind of like a red rag to a bull.
It's when he said basically: "Well things are going very well, we have all the information, and they [Democrats] have none of it."
Well, if you want a fair trial, then maybe that information should be made available.
We keep using the word "trial", and the words "jurors" and "witnesses" and "evidence", but we must not lose sight that this is a political process.
We saw that clearly last night when the first votes started coming in. In a vote that split completely along party lines, 53 Republicans said "no we should not be able to subpoena the White House for documents", while 47 Democrats said "yes we should".
So we have Donald Trump kind of goading and saying: "Look I've got the information. We know what happened, but we're not going to tell you."
I think this might inflame public opinion. Polls are already indicating that a clear majority believe that evidence should be handed over and witnesses should be called.
What is the witness swap proposal?
Democrats want to call Mr Bolton, who referred to the White House's alleged political pressure on Ukraine as a "drug deal", according to previous witness testimony in the House of Representatives.
But the former national security adviser has said he will not consider testifying unless served with a legal summons known as a subpoena.
Mr Trump's Republican allies have argued Hunter Biden should also be ordered to appear before the impeachment trial.
But Senator Chuck Schumer of New York, the top Democrat in the Senate, told reporters during a break in the trial on Wednesday: "That trade is not on the table."
Joe Biden said on Wednesday in Osage, Iowa, where he is campaigning for the White House that he would not offer himself up in any witness trade.
"We're not going to turn it into a farce or political theatre," Mr Biden said. "I want no part of that."
Defending his son, Mr Biden added: "There's nobody that's indicated there's a single solitary thing he did that was inappropriate or wrong - other than the appearance. It looked bad that he was there."
Mr Biden said last year that if elected president, no-one in his family would hold a job or have a business relationship with a foreign corporation.
What happened in the trial?
On Wednesday, the lead Democratic prosecutor, California congressman Adam Schiff, criticised President Trump's dealings with Ukraine as "worse than crazy".
"It's repulsive, it's repugnant. It breaks our word. And to do it in the name of these corrupt investigations is also contrary to everything we espouse around the world," he said.
Mr Schiff, who is chairman of the House Intelligence Committee, urged Republicans to vote to remove Mr Trump from office to "protect our democracy".
He warned that senators would "also undermine our global standing" if they do not oust the president.
The first day of the trial dragged on till the early hours of Wednesday morning as the senators debated a flurry of incremental motions.
Much of the evidence being laid out is a rehash of testimony already presented exhaustively in the House of Representatives, which voted to impeach Mr Trump last month.
Under arcane rules, senators are forbidden to drink coffee on the chamber floor and are only allowed water and milk.
A number of senators, mostly Republicans, were seen to be absent from the chamber during Mr Schiff's presentation.
Also, several members were spotted dozing during the proceedings.
What questions do you have about Donald Trump's impeachment trial?
Use this form to ask your question: