Mr Lawwell confirmed that for the last two years the club has been conducting its own investigations, with a "wholly independent and experienced lawyer" leading the work.
That investigation is still ongoing, with Mr Lawwell claiming criticisms of the club were "unfair and misguided"
Glasgow MSPs James Dornan and Adam Tomkins have both questioned the club's approach to the abuse scandal after former boys club coach James McCafferty was jailed for more than six years.
Mr Dornan, the SNP MSP for Glasgow Cathcart and a Celtic fan had accused Mr Lawwell and the club of continuing to "abrogate your responsibilities" after seeing the "latest in a far too long line of Celtic Boys Club coaches being found guilty of the foulest abuses".
He added: "In my view, you bring nothing but shame to the reputation of Celtic and the wider Celtic family."
Conservative MSP Mr Tomkins had called for a compensation package to be set up for those who suffered abuse, saying: "These are no longer allegations - in the cases in question they have been proved in a court of law."
Mr Lawwell has written to both MSPs, saying it is "important that we clarify a number of issues which appear to be misconceptions at present".
And he told Mr Dornan: "We believe that your criticisms, which suggested that we were not a caring club and that we were not taking our responsibilities seriously was both unfair and misguided".
McCafferty, 73, of Lisburn, Northern Ireland, was last month jailed for six years and nine months after pleading guilty to 11 charges related to paedophile activity against 10 victims over several decades.
That followed the jailing in November last year of James Torbett, who had been involved in setting up Celtic Boys Club, for what the court described as "depraved conduct" against three boys over an eight-year period in the 1980s and 1990s.
In February former youth football manager Frank Cairney was jailed for four years after being found guilty of historical sex offences from his time running a church youth team and, later, Celtic Boys Club
In the letters released to the Press Association, Mr Lawwell stated: "The first misconception is that the club is doing nothing and abdicating responsibility. That is simply not true."
Instead he claimed legal processes meant the club was "constrained" in what it could say publicly, describing it as being "highly frustrating for all".
But he also insisted it was not appropriate to discuss sensitive legal matters "through newspapers or on social media".
The Celtic chief executive said: "Some time ago our insurers appointed a wholly independent and experienced lawyer who is investigating and dealing with this matter on behalf of the club.
"It would be quite inappropriate for us to be discussing highly sensitive and confidential legal matters in the media, and we will not do so, even if that means we come under criticism for following the due legal process.
"We respect any claimants' rights and out advisers will communicate with them and their representatives directly in the proper manner, respecting their rights to confidentiality."
Celtic will "ensure that we continue to meet all our obligations", Mr Lawwell stressed.
Mr Dornan's letter had "suggested that the club was not caring, and the overall inference was that we were acting irresponsibly", Mr Lawwell said, adding that these claims were "both unfair and misguided"
The chief executive also claimed that in the "very delicate and of course tragic set of circumstances" the SNP MSP's letter had "appeared to disregard the importance of the due process of law".
He added: "Unfortunately legal processes are slow, and are also generally confidential. We have had to balance all of these factors in how was have addressed the issues to date.
"While we recognise that this issue is in the public domain we do not consider that means that we should deal with the matter through the media, but rather through the legal system.
"We would stress that we regret that the incidents took place and reiterate our sympathy for all victims who suffered abuse. We are following legal advice and respecting an ongoing process.
"The matter continues to receive our full attention and that we take all our obligations, including legal, very seriously."