Yulia Solodyankina, 22, who was studying physics at Edinburgh University, was reported missing on June 12.
She was last spotted on CCTV at Glasgow's Buchanan Street bus station at around 4.55pm on June 7.
Human remains were found near Arrochar in Argyll and Bute on Thursday.
No formal identification has taken place but it is understood the discovery is being linked to the search for Ms Solodyankina.
Last summer police said she may have travelled to the north or far west of Scotland.
Her parents made an emotional appeal after she disappeared, describing their daughter as "our life".
Based on previous trips she had made, detectives believed she boarded a bus in Buchanan Street, with the most likely destination the Highlands or Argyll and Bute.
A police spokeswoman said: "Police Scotland can confirm human remains were found in a rural location near to Arrochar on Thursday January 9.
"Further examination of the remains will take place.
"The inquiry is at a very early stage and nothing further is available at this time."