His comments follow an inquest which found that Mr Lawrence, 18, had been unlawfully killed four years ago "in a completely unprovoked racist attack by five white youths".
The Daily Mail provoked a general furore and outrage among many leading legal figures after it printed the names and photographs of five unconvicted young white men and accused them of being the murderers.
Three of the five, from south-east London, cannot be tried for Mr Lawrence's murder as they have already been acquitted and under British law cannot face the same charge.
Speaking at Commons question time, John Major told MPs: "I hope even at this stage that it will be possible to mount a successful prosecution. If evidence is forthcoming, it will be examined. There is no lack of will to prosecute."
He is understood to have been making a general comment about his desire for justice to be done, rather than talking about any specific suspects.
His comments came after Mr Lawrence's mother, Doreen, speaking yesterday at the launch of the 1997 European Year Against Racism, urged people to join the campaign to bring his killers to trial.
The Prime Minister went on to say the Daily Mail was cleared of any statutory contempt after it accused the five white men of Mr Lawrence's murder.
He said: "I'm not myself a lawyer, but I understand there's no question of statutory contempt as a result of the activities of the Daily Mail."Reuse content