Republican Mr McConnell filed a petition for a ‘cloture vote’, which if successful would limit debate on the Kavanaugh nomination and set a final 30-hour waiting period before a vote.
After filing a ‘cloture’, lawmakers must wait one legislative day before proceeding to the closure vote, which means Friday is the earliest it could take place.
Mr McConnell said on the Senate floor yesterday: “It’s time to put this sickening display behind us. The Senate will vote on this nomination this week.”
Judge Kavanaugh was nominated to the Supreme Court by President Donald Trump, but his appointment must be confirmed by the Senate.
Following his nomination, allegations of sexual assault emerged leading to an investigation whereby both his accuser, Christine Blasey Ford, and Mr Kavanaugh gave their testimony before the Senate Judiciary Committee
The FBI then investigated the allegations, focusing on high school friends who are supposed to have attended the 1982 party where the alleged assault took place.
The FBI report on the allegations neared completion yesterday and the Senate are expected to receive the report early today, according to Reuters sources.
Several people with information related to the allegations said they have not heard from the FBI, suggesting its report may be narrower than was desired by some of the lawmakers who demanded it two days ago.
Dr Ford’s lawyers say she has also not been contacted.
Three senators likely to play a pivotal role in the confirmation vote are Senators Jeff Flake, Susan Collins and Lisa Murkowski.
All three criticised Trump for his mocking remarks about Dr Ford at a Make America Great Again rally on Tuesday.
The Senate has 51 Republicans and 49 Democrats, and only 50 votes are required to confirm Kavanaugh, because Vice President Mike Pence would cast the tie-breaking vote.
This means the knife edge vote requires at least two Republicans to vote against his nomination for it to not go through.
In his testimony before the Senate committee, Mr Kavanaugh called the allegations “crazy stuff” and accused Democrats of a politically-motivated take-down of him.
Dr Ford gave an emotional account of her assault and claimed she is “100 percent” certain Judge Kavanaugh was her attacker.