Mr Finucane, 39, was shot at his home in Northern Ireland by loyalist paramilitaries, and his family has been trying to establish the scale of collusion by British security forces for the past 31 years.
A group of 24 members of US Congress – both Republican and Democrat – have written to the prime minister to accuse the British government of a “breach of faith” over the case.
“The British government’s deliberate decision not to proceed with an independent, judicial inquiry into the Finucane murder – an unfulfilled commitment of the peace process – is a public breach of faith,” said Republican congressman Chris Smith.
“It resonates throughout Northern Ireland and among international lawmakers and human rights observers, undercutting the good the British government has done and its commitment to justice, truth and reconciliation.”
Amnesty International has meanwhile written to Brandon Lewis, secretary of state for Northern Ireland, urging him to establish a public inquiry ahead of looming deadline.
In October Mr Lewis promised to make a decision by the end of November on whether or not an inquiry would proceed.
Mr Finucane, who was a prominent civil rights lawyer, was shot 14 times by loyalist paramilitaries at his north Belfast home on 12 February, 1989. The killing took place in front of his wife and three children.
In 2012 a government-commissioned review found that British security agents had been involved in the killing, but also said there had been “no overarching state conspiracy”.
However, last February the UK’s Supreme Court ruled that investigations into the murder had not been effective and fell short of international human rights standards.
The court did not order a new public inquiry be carried out, returning responsibility for the decision to the government.
“The government’s failure, after 31 years, to establish an independent judicial inquiry into the full circumstances of the murder, fuel the perception of a continued cover-up of the full extent of official involvement in the killing,” said Amnesty International’s director Kate Allen.
It has also emerged that Ireland’s premier Micheal Martin has contacted Mr Johnson about the case, adding his voice to calls for a public inquiry.
The late civil rights lawyer’s son John Finucane, the Sinn Fein MP for North Belfast, told an Irish parliamentary committee on Thursday that Mr Martin was “unequivocal and unambiguous” in giving the family the support of the Irish government.
Mr Finucane said: “The British government needs to heed and read the room with where this issue is going. It’s not going away.”
“We are no longer interested in who pulled the trigger – we want to know who pulled the strings and it is now time the British government live up to the promises they previously made.”
Mr Finucane also welcomed the intervention by the members of US Congress. “My family and I strongly welcome their continued support,” he said.