Brown, 21, was placed in a medically induced coma and the status of her brain function won't be known until the sedatives are reduced, the source told CNN.
"Bobbi Kristina is fighting for her life and is surrounded by immediate family. As her father already stated, we are asking you to honor our request for privacy during this difficult time," the Houston family said, thanking her supporters.
The source close to the family told CNN legal analyst Sunny Hostin that Brown's eyes opened and closed a few times Monday, but that doctors told family members not to read too much into that movement.
Doctors were trying to reduce the sedatives to check Brown's brain function, but decided not to and to keep her in the coma for now, the source said.
The source added that Brown had some seizures on Monday.
She "was not breathing" when a man identified as her husband found her Saturday morning inside her townhouse in the Atlanta suburb.
He called 911 and performed CPR until emergency crews arrived and took over, said police spokeswoman Lisa Holland.
The spokeswoman said when Brown was found, she was "unresponsive, meaning not breathing, no heartbeat."
In scanner traffic, emergency services could be heard describing an "ECHO-level response" -- urgent and life threatening -- with "possible cardiac arrest."
"Twenty-one-year-old female in the bathtub, facedown. PD's (police department's) en route."
Brown was taken to a local hospital, where "she is still alive and breathing. Other than that, I don't know her condition," Holland said over the weekend.
A source close to the family told CNN contributor Nischelle Turner on Monday that Brown is in "really bad shape."
Everyone is "hoping for the best, preparing for the worst," and "praying for a miracle," the source said.
Nobody knows what caused Brown's unresponsiveness, Holland said, but police consider it a medical incident at this time. She said investigators had found nothing to indicate it was drug- or alcohol-related. The incident report referenced a drowning.
Police had been to Bobbi Kristina Brown's residence recently. Somebody reported a fight there January 23, but nobody answered the door and officers found no evidence of an altercation, Holland said.
Almost three years ago, Whitney Houston was found dead in a bathtub at the Beverly Hilton hotel near Los Angeles, hours before she was to attend a pre-Grammy Awards party. A coroner ruled her February 11, 2012, death an accidental drowning
, with heart disease and cocaine use listed as contributing factors.
Brown is her only child.
"She encourages me and inspires me," Houston once said of her daughter. "When I look at her eyes and I see myself, I go, 'OK. I can do this. I can do this.'"
Daughter of acclaimed singer, R&B standout
Brown was born in 1993 during Houston's marriage to R&B singer Bobby Brown, which ended in divorce in 2007.
The daughter of music royalty became a public figure during the mid-2000s reality show "Being Bobby Brown," in which she frequently appeared alongside her parents and often had a front-row seat to their marital fireworks.
Bobby Brown requested privacy in a statement issued Sunday by his lawyer, who said that Brown was at the hospital with his daughter.
"Please allow for my family to deal with this matter and give my daughter the love and support she needs at this time," Bobby Brown said.
Mother and daughter performed together on national TV in 2009, when the two sang "My Love Is Your Love" in Central Park on ABC's "Good Morning America."
Given the pair's close relationship, it's no surprise that Houston's death was a major blow.
Days after her mother's body was found, a grieving Bobbi Kristina -- then 18 -- was taken to Cedars-Sinai Medical Center because she was "overwhelmed," family friend Kim Burrell said.
"She loves hard," Burrell said then of Bobbi Kristina. "She's going to be OK, and we're going to see to it."
After Houston's death, Bobbi Kristina talked of pursuing a career in acting and music, just like her mother.
In 2012, she reportedly lobbied for a role
in a planned biopic of her mother's life. Lifetime's "Whitney," which was directed by actress Angela Bassett, debuted in January 2015, but the Houston family was not involved.
Brown did make her acting debut in 2012 on Tyler Perry's TBS show "For Better or Worse." Her performance in the small role won praise from Perry.
"She did a fantastic job," he said at the time.
"And that kid has a such a future. She's such an actor. "
Even before her famous mother's death, videos of Brown singing surfaced on You Tube and were quickly criticized by fans who felt her talent did not live up to Houston's.
Concerns that she may have substance abuse issues, as her mother did, intensified in July 2014 after a picture was posted on her Instagram account
showing a young woman who looked like Brown appearing to be smoking from a bong.
The caption read: "This picture may be inappropriate but I want to make a connection with you all. Don't worry this picture isn't recent just hear me out. Do you know what I feel? I feel strange I can't feel any pain anymore. I don't take any drugs as of recently. Things just happened. But then there is some situation that force me to do things and the impact I'll feel pain. I did and I do. I was hurt."
Brown later denied that she posted the photo and tweeted that it was done by someone attempting to tarnish her image.
Married a man who was taken in by Houston
Brown appeared on episodes of Lifetime's brief reality show "The Houstons: On Our Own," which followed her life after Houston's death.
One of the show's storylines revolved around Brown's relationship with Nick Gordon, who had lived with Houston and her daughter since he was 12 years old.
Brown reportedly got engaged to Gordon
, and in January 2014, she tweeted out a picture of wedding rings, presumably belonging to her and Gordon, along with the words, #HappilyMarried• SO#Inlove."
"(Houston) made me promise several times to look after Krissy ... and, Mom, I will never ever, ever break that promise," Gordon told the entertainment show "Extra
" afterward, saying Houston "just treated me like she gave birth to me."
It was unclear whether Gordon came home Saturday morning to find Brown in the bathtub or if he had been in another part of the house overnight, Holland said.
Police have obtained a search warrant to look through the house, but that's standard procedure, she said.
Concern for Brown surged again after news broke.
Music director Michael Bearden
sent along "healing energy to the daughter of a late great friend." Songwriter Diane Warren tweeted
, "I hope Bobbi Kristina gets the help she needs."
"Poor thing," Warren wrote. "Almost 3 yrs to the day her mom passed and the same thing almost happened to her."