Lawmakers debated through the night amid last-ditch efforts by Israel's President Isaac Herzog to reach a compromise. He met Prime Minister Benjamin Netanyahu at a hospital where he had undergone surgery to fit a pacemaker.
On Monday, Netanyahu was discharged from hospital following the surgery, said Sheba Medical Center, where he was fitted with pacemaker.
Netanyahu had said from his hospital room on Sunday that upon his release he would go to parliament for the vote.
The proposed judicial revamp has split the nation and, since its unveiling in January, set off one of the biggest protest movements in Israel's history.
Protesters took to the streets again on Monday. Many gathered near parliament, where some camped out, to oppose Netanyahu's proposal to curb the powers of the Supreme Court as part of the revamp.
Critics fear the changes will undermine Israel's democracy.
Thousands of demonstrators backing the government and its reform plans had also rallied in Tel Aviv, the epicentre of months-long anti-government protests.
A vote is expected in the Knesset later on Monday on the draft law which would limit judges' ability to strike down government decisions they deem "unreasonable".
As the crisis looked set to come to a head, Biden urged Israeli leaders to postpone the vote.
"From the perspective of Israel's friends in the United States, it looks like the current judicial reform proposal is becoming more divisive, not less," he said in a statement first published by news site Axios and later shared with AFP.
"It doesn't make sense for Israeli leaders to rush this - the focus should be on pulling people together and finding consensus."
'Time of emergency'
Herzog, who returned from a visit to the United States on Sunday, plunged directly into arriving at a compromise, also meeting opposition leader Yair Lapid.
"This is a time of emergency. An agreement must be reached," Herzog said in a statement issued by his office.
His office declined to comment on his talks with Netanyahu and Lapid.
The announcement from Netanyahu's office that the 73-year-old was having a pacemaker fitted came days after he had been hospitalised for a reported spell of dizziness.
On Sunday afternoon, Netanyahu issued a video saying he was "doing great."
"We're continuing our efforts to complete the legislation, and the efforts to do it in agreement (with the opposition)," he said.
"Either way, I want you to know that tomorrow (Monday) morning I'm joining my friends at parliament," he said.
The driving force behind the reforms, Justice Minister Yariv Levin, said the bill being put to lawmakers had already undergone changes to accommodate critics, but added that the coalition was still open to "understandings".
"Understandings means the opposition's willingness to make concessions too," he told supporters at the Tel Aviv rally on Sunday.
Netanyahu's government, which includes far-right and ultra-Orthodox Jewish allies, argues that the proposed reforms will prevent overreach by unelected judges and ensure a better balance of power.
'We can still stop'
Opponents accuse Netanyahu, who has been fighting corruption charges in court, of a conflict of interest and some protesters have labelled him the "crime minister".
"We have to keep up the pressure, we have to safeguard our democracy," said one demonstrator, Amir Goldstein.
Inside the chamber on Sunday, Lapid echoed such concerns.
"We want to continue to live in a Jewish and a democratic state ... We must stop this legislation," he said.
Opposition leader Benny Gantz issued a similar call.
"We can still stop, come to an agreement on the reasonability clause," he told the Knesset. "We have to stop everything."
If approved, the "reasonability" clause would be the first major component of the reforms to become law.
Other proposed changes include allowing the government a greater say in the appointment of judges.
The protests have drawn support from across the political and social spectrum, among secular and religious groups, peace activists and military reservists, blue-collar and tech sector workers.
One protester opposed to the judicial reform package, Shanna Orlik, said she was rallying against what she called a "misogynist and far-right government".
"We don't have a constitution, and the only thing that protects our rights is the Supreme Court, and the government intends to destroy that," she said.