While recent discussion in the fashion world has centered on the need to support new designers, rather than revive older names, Mr. Idol said Kors was not interested in emerging faces, but rather those with “some longevity” that “may need to have a structure to accelerate their growth.”

At the same time, he added, Kors was attracted to businesses where a founder or family member remained involved.

“It’s become harder and harder for independent designers to make the leap to household name,” said Robert Burke, the founder of a luxury consultancy that bears his name.

Coach is a little further along. It began life as an accessories business, unlike Kors, which is known for its ready-to-wear items as much as its signature tote. In addition to Kate Spade, a label that operates at the more affordable end of the retail spectrum, it has bought the upmarket shoe brand Stuart Weitzman.

Mr. Luis has also been open about his ambitions for more acquisitions. Indeed, Coach was rumored to be a suitor for Jimmy Choo, underscoring the potential that Michael Kors and Coach will pursue similar targets.

A number of possible acquisitions for both companies have already been mentioned, many of them hailing from across the Atlantic. Analysts have pointed to accessories brands like Furla in Italy and Longchamp in France; both are family-owned businesses with strong records of growth and established consumer appeal across continents and generations.