The Arizona Republican is retiring at the end of this year.
Activists plan on rallying outside of City Hall Plaza starting at 10 a.m., calling for Flake to reject Kavanaugh's nomination.
The White House has denied reports it is trying to limit the FBI investigation into the president's Supreme Court pick.
"The White House and the FBI should release details on the scope of the Kavanaugh background investigation and keep Congress informed on any changes", Senator Dianne Feinstein, the committee's ranking Democrat, tweeted on Sunday.
U.S. president Donald Trump, for his part, tweeted that no matter how much time and discretion the Federal Bureau of Investigation was given, "it will never be enough" for Democrats trying to keep Mr Kavanaugh off the bench.
Kavanaugh has denied Ramirez's allegation. According to a person familiar with the matter, she detailed her allegation that Kavanaugh exposed himself to her at a party in the early 1980s when they were students at Yale.
Trump on Friday had ordered the new FBI investigation into sexual assault allegations against Kavanaugh, as the Senate delayed a final vote on his confirmation as the key ninth member of the Supreme Court.
Kavanaugh has denied assaulting Ford.
Mr Ludington said he is "deeply troubled by what has been a blatant mischaracterisation by Brett himself of his drinking at Yale".
While saying that youthful drinking should not condemn a person for life, Mr Ludington said he was concerned about Mr Kavanaugh's statements under oath before the Senate Judiciary Committee.
"We're staying out of the way", White House press secretary Sarah Sanders told "Fox News Sunday". We're not micromanaging this process.
"It's a Senate process".
Coons and Flake met in a tiny hallway outside the Senate meeting, and as other members of the committee got involved in the discussions, the two squeezed into "literally a phone booth that we can barely fit in" to continue discussions with another senator, Coons said. "It's not meant to be a fishing expedition", White House adviser Kellyanne Conway said on CNN's "State of the Union". "The FBI is not tasked to do that".
But Democratic members of the Senate Judiciary Committee appear unconvinced.
Coons had been talking to friends and colleagues about taking a "one week pause, one week only".
"I think that will be the scope of it".
In response to her testimony, Mr Kavanaugh said he had never assaulted her or anyone else.
NBC News, the New York Times and the Wall Street Journal previously reported that the White House was constraining the investigation, prompting Democrats on the Senate Judiciary Committee to express concern. In a letter Sunday, she also asked for updates on any expansion of the original directive.
A White House official, speaking on condition of anonymity, told Reuters the White House had defined the parameters of the probe for the FBI and that the investigation would start with interviews with only four people.
Who are Mr Kavanaugh's accusers?
Collins and Flake spoke throughout the weekend.
In a separate report, NBC quoted a senior USA official and a second source as saying the limits imposed by the White House on the inquiry remained in place, despite Trump's tweet.
Jeff Flake had a message for his friend Democratic Sen. Lisa Murkowski of Alaska, said McConnell's spokesman Don Stewart.
"I thought it was going to be an investigation, but instead it seems it's just an alibi for Republicans to vote for Kavanaugh", he said.
President Trump ordered the inquiry on Friday at the request of a Senate committee. "We have to allow them to go ahead". "And hopefully at the conclusion everything will be fine". Kavanaugh has called her accusations a "joke". He has vehemently denied all allegations.
The first woman to come forward with allegations against Mr Kavanaugh was psychology professor Christine Blasey Ford. Judge has also denied misconduct allegations.