A Montana ranch featured in the 1992 Robert Redford movie “A River Runs Through It” is in contract, less than a week after the listing was made public, the listing agent confirmed. The property, known as the Climbing Arrow Ranch, sold for around its $136.25 million asking price following a bidding war, according to two other people familiar with the deal.

“Since our public release of the listing of the Climbing Arrow Ranch, we have received an extraordinary response from the market, with multiple parties stepping forth immediately,” listing agent Mike Swan of Swan Land Company said in a statement. “We can confirm that the sellers have engaged with a prospective buyer who is currently conducting their inspections of the ranch.”

The closing is slated to happen sometime this summer, according to a spokeswoman for Mr. Swan.

The sellers are the Anderson family. They have owned the property since 1959, when it was purchased by Buck Anderson, grandson of Bank of California President Frank B. Anderson, and his wife Marcia Anderson, according to the agent. The property has been expanded over time.

Mr. Anderson died in 2012, and Ms. Anderson died last year. The property is being sold by their children. The family could not immediately be reached for comment.

Located roughly 30 miles from Bozeman and spanning four counties, the ranch comprises a series of five separate, noncontiguous parcels that total about 80,000 acres, one of which borders ranches owned by media mogul and prominent landowner Ted Turner, Mr. Swan said. The ranch includes irrigated hay meadows along the Madison River and limestone cliffs. It is also known for its bull-elk hunting and trout fishing.

The property includes a herd of nearly 2,000 commercial Black Angus cattle and an abandoned railroad, from the days that the Milwaukee Road ran through a portion of the property. There are seven modest ranch homes on the ranch, most of which are for staff.