French, who has dual nationality, and his Norwegian friend Tjostolv Moland, were sentenced to death for murder and spying in the vast central African country in 2010.
In a letter sent by London-based charity Reprieve, which campaigns against the death penalty, French's mother Kari Hilde urged British Prime Minister David Cameron to ask DR Congo's President Joseph Kabila for clemency.
"The decision whether or not to take Joshua's life is in the hands of President Kabila," Hilde said in a statement published by Reprieve.
"It is now essential that both the Norwegian and the British Prime Minister become involved and ask, on behalf of their nations, that these young men be spared," she added.
"I request of the British prime minister to ask His Excellency President Kabila for a full pardon for my son."
Former soldiers French, 29, and Moland, 30, were first sentenced to death in 2009, convicted of having killed the driver of the car they had rented in May that year, but they insisted the driver was killed by bandits.
The verdict was annulled for irregularities by the high military court in 2010, and new judges heard a retrial.
French and Moland, who met in the Norwegian army after French trained as a British paratrooper, were then retried and convicted in June 2010 of spying, criminal association, murder and attempted murder.
A prison official claimed in August last year that the pair had tried to escape, but their lawyer denied this.
A spokesperson for Britain's Foreign Office told AFP the British government was in regular contact with French's family and had raised the case regularly with Congolese authorities.
"The British government is absolutely opposed to the death penalty in all circumstances and we are committed to appropriate influence to prevent the execution of any British national," he said.
DR Congo has not carried out a death sentence since Kabila came to power in 2001, but has commuted such penalties to life imprisonment.