Mr Johnson, 57, told the Mirror on Friday that the club paid him £50,000 to keep quiet about allegations of sexual abuse by a former chief scout.
In a statement, Chelsea said Mr Johnson had "suffered unacceptably".
The club said a review into the case would take place, adding it had "no desire to hide any historic abuse".
It said the review would examine whether it had carried out a proper investigation when the allegations first came to light and why it did not report them to the Football Association and Premier League.
"We are fully committed to ensuring the safety and well-being of all children and young people who are in our care or attending our premises.
"Their welfare is of paramount importance," the club said.
Mr Johnson was a member of Chelsea's first team from 1978 to 1981.
He joined the club as an 11-year-old in 1970 and told the Mirror he had been groomed from the age of 13 by scout Eddie Heath.
Mr Heath, who was the club's chief scout from 1968 to 1979, died before the allegations against him were made.
According to the Mirror, Mr Johnson signed a confidentiality agreement in 2015 and accepted £50,000 from the club, but they did not accept blame.
Chelsea said in its statement that when the settlement was reached the club's board understood it was "usual practice" to include a mutual confidentiality agreement, adding that Mr Johnson's solicitors had not objected to the clause.
"More recently, against the current backdrop of wider revelations and other victims coming forward bravely to tell their story, we no longer felt it appropriate to keep the confidentiality agreement in place. It was therefore removed," the club said.
The club said the decision to have a confidentiality clause had received "significant scrutiny" and it had now asked an external law firm to review this decision and make recommendations for settling claims in the future.
"In advance of that, however, the board would like to make clear that, in light of what we know now about the wide-scale abuse in football clubs in the 1970s and 1980s, it now believes that the use of such a clause, while understandable, was inappropriate in this instance.
"We certainly have no desire to hide any historic abuse we uncover from view. Quite the opposite."
Chelsea said Mr Johnson's solicitors at the time claimed Heath had "inappropriate relationships" with other young boys and men from the club, but no names were given.
The club said with the "limited information" it received it had been unable to identify any further individuals who may have been subject to abuse.
It comes after former Chelsea midfielder Alan Hudson said on Facebook that it was "common knowledge that Mr Heath was a danger to us youngsters".
Chelsea added it was now a "very different club" to the one it was in the 1970s.
It said its safeguarding policies were robust and continually reviewed and audited by the Premier League.
Meanwhile, an ex-Southampton FC employee who has been accused of abusing young boys at the club in the 1980s is still working in football, the BBC understands.
The man, who has not been named for legal reasons, is understood to have left Southampton after concerns were raised about his behaviour - but is now employed by another football club.
Some 18 police forces have announced investigations into claims of sexual abuse in football.
At least 350 people have come forward alleging they were victims and 55 football clubs are linked to allegations of abuse.
A dedicated NSPCC helpline received more than 800 phone calls in a week. The line is available 24 hours a day on 0800 023 2642.
Police forces investigating allegations of historical sexual abuse within football:
Kent Police also said it had received reports which it was reviewing.
Chelsea Football Club has apologised "profusely" to former footballer Gary Johnson over sex abuse he suffered as a youth team member in the 1970s.