Once completed, it is intended that The Studio will return to its original use as a design studio.
It will also provide a permanent base in the Borders for the BKF and its exhibitions, workshops and talks.
Its chair of trustees, Prof Alison Harley, said these were “exciting times” for the organisation.
“Our new partnership will be focusing on the entwined legacies of Bernat Klein and Peter Womersley and this unique opportunity for the creative industries,” she said.
“In this next important stage of the project, the BKF will continue to support new creative output through its public programme inspired by Klein’s creative and cultural legacy.”