EAST HAMPTON, N.Y. — The client wanted a 10-day house rental in the Hamptons that offered privacy and proximity to East Hampton Airport. Several important guests would be visiting.

The real estate broker handling the account lined up three options. The first one she showed the client’s manager was a farmhouse built in the 1800s. The door frames were deemed too low.

“We would cut to the chase much better if I knew who it was,” said the broker, Renee Gallanti, recalling how she was not told at first who the client was for that house-hunting mission in 2016. “At first they tried to give very little information at all. And after we got to the first house and the ceilings were too low, we shifted gears.”

The client, she soon learned, was Kevin Durant, the 7-foot N.B.A. All-Star. He was about to become a free agent for the first time — and would soon find himself at the heart of a courtship unlike any the N.B.A. had ever seen.