Police Service of Northern Ireland (PSNI) and Police Ombudsman investigations into the 1989 loyalist paramilitary shooting are to go ahead.
Mr Finucane, a 39-year-old solicitor who represented republican and loyalist paramilitaries during the Troubles, was shot dead in his family home in north Belfast in February 1989 by the Ulster Defence Association (UDA) in an attack found to have involved collusion with the state.
Mr Finucane’s widow Geraldine and the couple’s three children have been campaigning for decades for a public inquiry to establish the extent of security force involvement.
Last year, the Supreme Court said all previous examinations of the death had not been compliant with human rights standards.
Northern Ireland Secretary Brandon Lewis said he met the Finucane family on Monday.
“I advised them of my decision not to establish a public inquiry at this time,” he said.
“It is in the public interest to allow the police and ombudsman processes to proceed before taking any decision on whether the state’s Article 2 (duty to investigate deaths) obligations have been discharged or whether further steps are required.”
Mr Lewis reiterated previous British Government apologies over the killing.
PSNI Chief Constable Simon Byrne said: “It is our view that there are currently no new lines of inquiry.
“We now need to decide if a further review is merited given all the previous investigations into this case.
“Once we have determined that, we will inform the Finucane family.
“If we determine that a review should take place, we will then have to decide if we are best placed to carry out that review.
“As it stands it is unlikely that we would enjoy a perception of independence in this case, given the accepted position of state involvement in this matter. Therefore, it is highly likely that any review would need to be conducted independently.”
The Supreme Court acknowledged Mrs Finucane had been given an “unequivocal undertaking” by the Government following the 2001 Weston Park agreement that there would be a public inquiry into the murder.
However, the judges found that the Government had been justified in later deciding against holding one.
The court said it was up to the Government to decide what form of investigation was now required, if one was feasible.
Amid a Government delay in responding to the judgment, Mrs Finucane took fresh judicial review proceedings against the state.
Reporting by PA