Officers working on a missing person inquiry made the discovery in Market Street at about 23:55 on Wednesday.
A man has been detained and is helping police with their inquiries.
A Police Scotland spokesman said: "The investigation into the circumstances is in the very early stages of establishing who the remains belong to and how that person died."
He added: "Our inquiries have taken us to a number of addresses in the Montrose area where there is likely to be a police presence for some considerable time."

The 1.5 percentage point increase was higher than expected, with analysts having forecast a rise of 0.5 percentage points.
The bank has already raised rates from 5.5% at the start of the year but the moves have failed to combat inflation.
A weak rouble and a ban on western food imports has kept inflation stubbornly high.
"If external conditions improve, and a persistent trend for lowering inflation and inflation expectations emerges, the Bank of Russia will be ready to start to ease its monetary policy," the central bank said.
The new rate will take effect on 5 November. The last rise was imposed at the end of July.
The central bank said that inflation had reached 8.4% and would remain above 8% until the end of March.
Economic growth is expected to almost grind to a halt in the final three months of this year and in the first quarter of 2015.
The rouble briefly firmed after the bank's decision was announced, but then fell back into negative territory.
The rise comes as Russia said it would will resume shipments of natural gas to Ukraine after Kiev makes its first payment for previous supplies next week.
Meanwhile, the European Union said it had launched a trade dispute with Russia at the World Trade Organisation (WTO) to challenge Russia's treatment of European agricultural and manufactured goods.
The WTO said the EU had accused Russia of levying higher-than-permitted tariffs on a range of goods including paper, palm oil and refrigerators.
The dispute is the fifth case involving Russia and the EU that has been brought to the WTO.

He is refusing to stand for the The Star-Spangled Banner in protest against the plight of black people in the US.
On Sunday Kaepernick, 28, threw his first touchdown pass in 364 days in the 49ers' 45-16 loss to the Bills.
"To be honest, I was focused on football," Kaepernick said.
"I really wasn't listening to what was going on in the stand," he added.
Some fans also chanted "U-S-A" before the national anthem at Buffalo's New Era Field, while t-shirts featuring the player's face and a rifle target were reportedly on sale close to the ground.
Kaepernick added: "I had some Bills fans come up before the game and say they support me."
The quarterback said he has received death threats as he continues his protest, which stems from racial tensions that increased across the nation this year amid a string of police killings and subsequent revenge attacks.
His protest has drawn strong reactions both in opposition and support, with some fans calling it disrespectful to the US flag.
But some veterans and military members have tweeted their support for Kaepernick with the hashtag #VeteransForKaepernick.
"At the end of the day, I'm going to continue to fight for what I'm fighting for," he said. "I think the message carries weight because people realise what's happening - it's something that needs to be addressed."
Asked about a claim that someone in the stands threw a bottle at him, Kaepernick replied: "If they did, they didn't have very good aim."
Police have found what are believed to be human remains at a house in Montrose.
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Russia's central bank has raised its key interest rate to 9.5% from 8% as it seeks to tackle inflation.
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San Francisco 49ers quarterback Colin Kaepernick, who has polarised opinion by kneeling during the national anthem, was booed on his return to the starting line-up against the Buffalo Bills.