The owners of the house where Marilyn Monroe last lived and died are suing the city of Los Angeles over what they call “backroom machinations” as part of efforts to landmark the house and save it from a planned demolition.

In a lawsuit filed in Superior Court in Los Angeles County on Monday, lawyers for Brinah Milstein and Roy Bank accused the city of violating its own codes and conspiring with third parties to secure its desired outcome during a hurried process to designate the house at 12305 Fifth Helena Drive a historical landmark last fall.

This lawsuit highlights how the city engaged in a “corrupt process to guarantee their preferred outcome rather than engaging in a neutral and fair process,” Peter C. Sheridan, a lawyer for the couple, said in a statement. The city did not respond to a request for comment.

Ms. Monroe was the world’s most famous woman when she moved in March 1962 to Fifth Helena Drive, a secluded residential street in the Brentwood neighborhood that is part of a set of 25 cul-de-sacs off Carmelina Avenue.

The actress became a pop culture icon in the 1950s with roles in movies like “All About Eve,” “The Seven Year Itch” and “Some Like It Hot.” But her time in Brentwood would not be long: In August 1962, Ms. Monroe died of a drug overdose in her bedroom at age 36. Fans and landmark preservationists have argued that the house is a part of Hollywood history and should be designated a protected property as a part of her legacy.

Though Ms. Monroe’s house is not visible from the street, tourists frequently stop to leave flowers or try to catch a glimpse of the home. The house became known as Cursum Perficio, which in Latin loosely translates to “I end the journey,” and is a Spanish Colonial-style property that is partly inlaid with ceramic tile.

The original house was believed to have been built in 1929, and most of the alterations to it were done before Ms. Monroe bought the property for $75,000, according to the city’s application for historic designation, “and therefore have gained significance as related to the period of her occupancy.”

“The subject property is the first and only residence Monroe ever purchased by herself, and represents a portion of her productive period and an embarkation on a new phase of her life,” the application reads.

Ms. Milstein, an heir to a wealthy real estate family, and Mr. Bank, a reality television producer, own the property next door and purchased the Monroe house last July for $8.35 million to combine the properties and expand their current home. They soon after applied for demolition permits.

The owners argue that “the house has been substantially altered since 1962,” the lawsuit says. “There is not a single piece of the house that includes any physical evidence that Ms. Monroe ever spent a day at the house, not a piece of furniture, not a paint chip, not a carpet, nothing.”

Over 60 years, 14 owners and numerous permitted remodels, “the city has taken no action regarding the now alleged ‘historic’ or ‘cultural’ status of the house,” the lawsuit states.

City records showed a demolition permit had been issued for the single-family home, attached garage, pool house and storage. Records also showed there were plans to backfill the kidney-shaped pool lined with palm trees, which was captured in photographs as police responded to the scene of Ms. Monroe’s death in 1962.

But in “a spasm of activity,” lawyers for the owners said in the suit, city staff and Councilwoman Traci Park, who oversees the district, “arranged the desired outcome” when the City Council voted unanimously to begin the designation process, triggering a temporary stay on the demolition permit that the city’s building department had approved just days prior.

Ms. Park did not return a request for comment.

Ms. Milstein and Mr. Bank have expressed concern that a historic designation would spur an increase in tourism on the small, private road. At a cultural heritage hearing in January, Ms. Milstein described tourists ringing their gates as well as their neighbors’ to be let into the house. Some offered money to take pictures because the house cannot be seen from the street.

A full City Council vote is expected this spring to formalize the designation. Now the owners are hoping to restore their right to demolish the property.

The couple has offered to relocate the house so the public can interact with the building, and even secured the backing of Authentic Brands Group, which controls Ms. Monroe’s estate and is a co-owner of Elvis Presley’s Graceland. The Brentwood Community Council, an organization that says it represents about 35,000 stakeholders including homeowner and business groups, and several other homeowners associations in the area oppose the designation and support relocating the house.