Mr Griffin began the final 60-mile trek on skis on Thursday and is due to reach the landmark by Thursday next week.
The boss of the minicab and courier firm was inspired to undertake the challenge by the quest of Tom Crean more than 100 years ago.
Crean, who was married to Mr Griffin’s great-grandfather’s sister, took part in three Antarctic expeditions more than 100 years ago, including Scott’s failure to arrive first at the South Pole.
Crean, who was left out of the final team, saved the life of a colleague, earning him the Albert Medal.
His third attempt to reach the pole was with Shackleton and ended with the crew spending more than a year adrift before reaching land.
Mr Griffin, who is accompanied by Ifan Jenkins and expedition leader Louis Rudd, an ex SAS captain, is braving temperatures as low as -50C, battering winds and steep ridges on the 12-day, 100 mile mission.
A keen adventurer, he has previously completed the Le Mans 24-hour race and the Dakar rally. “This journey has a personal significance for me – being able to finish what a member of my family started over 100 years ago,” he said.