The Jim Clark Trust wants to establish a dedicated museum housing cars, trophies and memorabilia and hopes the project will be completed in 2018.
Sir Jackie has agreed to be honorary president while Franchitti is also supporting the museum.
It would replace the existing Jim Clark Room in Duns, Berwickshire, which currently houses displays celebrating the driver who was world champion twice. He died in 1968,
Ian Calder, Clark's nephew, said: "Our aim is to keep his legacy alive with the new museum in Duns to inspire the next generation."