BBC Radio 4's File on 4 programme discovered 22 profiles belonging to men convicted of child sex offences, which included grooming a child on Facebook and attempting to meet her for sex.
Six of the accounts remained live three weeks after the BBC reported them.
Facebook's UK policy director Simon Milner said the company was effective in dealing with the issue.
He said he appreciated the BBC bringing the pages to light but said Facebook worked proactively to search out behaviour that should not be happening on the site.
According to the platform's own rules, convicted sex offenders are not allowed to use the social media network.
Among the profiles uncovered was one belonging to Bruce Cordwell, 20, who groomed 15-year-old Kayleigh Haywood through Facebook and Whatsapp just days before she was raped and murdered by another man, Stephen Beadman, in November 2015.
"It's absolutely vile," said Stephanie Haywood, Kayleigh's mother, when she was shown Cordwell's page.
Kayleigh, from Measham, Leicestershire, was contacted by 28-year-old Luke Harlow on Facebook in 2015 and the pair exchanged more than 2,600 messages during a two-week period.
After Harlow convinced art enthusiast Kayleigh he was her boyfriend, she agreed to stay over at his flat in Ibstock.
The following night, his neighbour Stephen Beadman came round to the flat and later raped and murdered the teenager before dumping her body in a hedgerow.
Beadman was jailed for 35 years for murder, rape and false imprisonment, and Harlow received a 12-year sentence for meeting a child following sexual grooming, sexual activity with a child and false imprisonment.
But while detectives were investigating the case, they discovered Cordwell, then 19, had also been sexually grooming Kayleigh.
In February, he was jailed for three years and seven months after pleading guilty to attempting to arrange the commission of a child sex offence.
Stephanie, a mother-of-eight, said convicted child sex offenders who remained on social media posed a huge risk.
"There is a risk they could do what they've done again, or worse," she said.
"I say, 'Once a groomer always a groomer.'
"Once they've been inside for whatever they've done, I think they should remove it. I think it's horrible."
By comparing police mug shots and media coverage of child sex abuse cases and cross-referencing them with the details on Facebook, File on 4 tracked down accounts belonging to 22 convicted offenders.
Facebook has a rule that forbids convicted sex offenders from using its site and has a portal for people to report pages belonging to offenders.
The portal requires people to supply a link to the offender's page and provide evidence of their offending, which could be a link to a media report of the court case.
Facebook then checks on the offenders with the police before removing them from the site.
When the BBC reported the 22 pages it had found, most were removed within 48 hours - but six remained live three weeks after the BBC reported them.
Facebook said it was waiting for confirmation from police that the final six were convicted offenders before it could remove the pages.
In response to File on 4's findings Facebook's UK policy director Simon Milner, said: "We appreciate you did that, we appreciate it when anybody reports to us things that are happening on Facebook that shouldn't be.
"Our teams that focus on this work with the relevant local police force to get authorisation from them and it works."
Mr Milner said Facebook worked proactively to search out behaviour that should not be happening on the site.
He added: "We've found in terms of our working relationship with law enforcement in this country and elsewhere they think we are amongst the most effective companies in dealing with this issue and we actually do have the right kind of arrangements in place."
Meanwhile, exclusive figures obtained by File on 4 through Freedom of Information requests revealed the most popular social media platforms had been linked to more than 7,000 reports of child sex crimes over the past three years.
Figures from 22 police forces revealed 7,162 child sex offences had been recorded between 2014-15 and 2016-17, with a 44% increase recorded over the three-year period.
The offences included rape, sexual activity with a child and child sexual grooming involving children as young as three years old.
File on 4: Online Grooming is on BBC Radio 4, 13 June at 20:00 BST - catch up on BBC iPlayer Radio.
Have you got something you want investigating? We want to hear from you. Tweet us, or email fileon4@bbc.co.uk
Facebook accounts belonging to convicted child sex offenders have been uncovered by a BBC investigation.