(CNN) Saudi Arabia is pushing back on reports that shortly before two Saudi sisters were found dead in New York it had ordered anyone related to them to return to the kingdom because they had sought political asylum.

Rotana Farea, 23, and Tala Farea, 16, were found dead October 24 along the banks of the Hudson River after their family last saw them nearly a year ago in Virginia. They had been in the United States for two to three years, police said.

Saudi officials are looking into whether the sisters applied for asylum in the United States, a Saudi official told CNN.

Earlier this week, The New York Times reported the Saudi Embassy in Washington had called the mother of the two sisters to inform her that her daughters had applied for asylum.

But Saudi Embassy spokeswoman Fatimah S. Baeshen disputed "reports that we ordered anyone related to the Saudi sisters, Tala and Rotana Farea, God rest their souls, (who recently died tragically in NY), to leave the US for seeking asylum."

Calling the claim "absolutely false," Baeshen said in a Twitter post Friday, "details are still under investigation and will be shared in due course."

Reports that we ordered anyone related to the Saudi sisters, Tala and Rotana Farea, God rest their souls, (who recently died tragically in NY), to leave the US for seeking asylum; are absolutely false. Details are still under investigation and will be shared in due course. — Fatimah S Baeshen (@FatimahSBaeshen) November 2, 2018

The sisters were Saudi citizens, and their nation's royal consulate general has been in contact with their family and has appointed a lawyer for the investigation.

The sisters had said they would rather kill themselves than return to Saudi Arabia, New York police said its detectives had learned. But so far, investigators have uncovered no evidence that the sisters were victims of foul play.

"At this point in time we have no credible information that a crime took place in New York City," Chief of Detectives Dermot Shea said Friday, "but it is still under investigation."

Shea said earlier this week that detectives have been in Virginia conducting interviews with members of the immediate family to shed light on "what was going on in the two young ladies' lives."

"We've made significant progress in piecing together pieces of this puzzle to find out what happened," he said.

The sisters were last seen by their family in Virginia on November 30, 2017, Shea said.

Before their disappearance, the sisters were living in some kind of "shelter-like" facility due to abuse allegations that came up. Shea did not elaborate or give further details on those reports, which he said had not been corroborated.

According to Shea, investigators believe the sisters left Fairfax, Virginia, on either August 23 or 24 and took different methods of transportation, including Uber, to New York, where they arrived on or about September 1.

Until their bodies were discovered, the sisters stayed at high-end hotels, went shopping and ordered food on a credit card until police believe "the money started to run out," Shea said. The credit card was maxed out.

In video police have seen, the sisters were in "apparently good health" as recently as a week before their bodies were found on the riverbank, duct-taped together in a way that appeared to be "keeping them together," Shea said, but was not meant to bind them.

Investigators have also spoken to an eyewitness who came forward Wednesday and told police he saw two girls sitting in a playground park adjacent to the Hudson the day the bodies were found.

The witness was out exercising at about 7 a.m. when he saw two girls sitting about 30 feet apart with their heads in their hands, praying.

Shea described the witness as a "credible source" who told police the encounter had been "haunting him."

Shea also confirmed investigators believe the girls were alive when they entered the water. The location where the witness saw two girls was "a very short distance from the Hudson River," and such that they could have walked right in.

Investigators suspect suicide, official says

Fairfax County police previously said the sisters were reported missing for one day in December.

Officer Tara Gerhard, a spokeswoman for the Fairfax department, said the sisters were "referred to services," but she could not specify what type of services, or offer additional details, citing privacy laws.

A law enforcement official who spoke to CNN on the condition of anonymity previously said investigators have not found anything to indicate the sisters were victims of foul play.

Shea echoed that belief in Friday's news conference, saying, "It is entirely credible that the girls entered the water alive."

Police previously said their bodies showed no signs of trauma. Shea dismissed reports they jumped from the George Washington Bridge because there were no body marks that would indicate a fall from that height.

The New York Police Department is not ready to determine the official causes of death. Authorities are investigating whether the deaths were the result of suicide, homicide or an accident, said Phil Walzak, deputy commissioner for public information for the New York police.