Durst owned a health food store called All Good Things in Middlebury at the same time 18-year-old Lynne Schulze was reported missing, police said.
"We are aware of the connection between Robert Durst and the disappearance of Lynne Schulze. We have been aware of this connection for several years and have been working with various outside agencies as we follow this lead," Middlebury police said in a statement today.
"This is an ongoing criminal investigation and the Middlebury Police Department is not releasing any other details," the statement continued.
Police are examining Durst's actions in connection with five deaths or disappearances in four states: two in California, one in New York, one in Texas and one in Vermont, according to law enforcement authorities.
Durst's attorneys have said Durst is innocent.
"Bob Durst didn't kill Susan Berman and he doesn't know who did," Dick DeGuerin, his attorney, said outside court last Tuesday.
Durst has denied killing Berman, but some believed he had a motive, allegedly wanting to silence her about the 1982 disappearance of his first wife, Kathleen Durst, in New York.
Kathleen Durst was officially declared dead in 2001, and Durst has said he has no idea what happened to her.
Durst is also being examined in connection with the 1997 disappearance of 16-year-old Karen Mitchell.
Mitchell was last seen at a mall in Eureka, California, police said. Durst was living in Trinidad, California at the time, which is 25 miles north of the mall.
Investigators never arrested anyone in connection with Mitchell's case, though they do believe she "met foul play."
"We are certainly interested in any information that may or may not come out of interviews with Mr. Durst," Eureka police chief Andy Mills told ABC affiliate KAEF. "If information comes to us that allows us to further our investigation we will certainly take the opportunity to do that."
Durst was also charged in the 2001 killing of a neighbor in Galveston, Texas, but he claimed self-defense and was later found not guilty.