The remains were found in Pasvik in the Sør-Varanger municipality, reports newspaper VG.
Kaisa Beddari, who was 85 years old when she disappeared in 2013, is the only person reported missing in the area, according to police.
Police in Finnmark wrote in a press statement that they would investigate whether the discovery is connected to the disappearance of the elderly woman.
Reports were received by police of the possible discovery of human remains on Friday, according to VG's report.
“We went out on Saturday and investigated the area, and engaged extra resources today in order to find more of the remains,” regional leader Hans Møllebakken of Kirkenes Police Station told the newspaper.
The police officer said that the remains were found in the Skrøytnesmyra wetland area around one kilometre from Beddari's home.
“There is basis to believe that we have found her,” he said.
When Beddari was reported missing in 2013, she was thought have gone out to pick berries, according to a report by Nordlys published five days after her disappearance.
She was seen carrying a basket for picking berries on the afternoon of her disappearance, according to the report.
“A long and comprehensive search for her was conducted,” Møllebakken told VG on Monday.
The search for Beddari was formerly called off in November 2013 and she was declared dead by Øst-Finnmark District Court in April 2014, writes VG.
Møllebakken added that it was too early to say anything about a possible cause of death.