Police Scotland said the emergency services were alerted to the incident at about 13:00.
A coastguard helicopter and lifeboat joined rescue teams looking for the man.
A post on the RNLI Montrose Lifeboat Facebook page said it had been stood down at 18:20 "due to failing light".

Kevin Nunes, 20, a drug dealer, was seized at gunpoint and shot dead in Staffordshire in September 2002.
The Crown Prosecution Service said it did "not seek to uphold the convictions" of the men nor press for a retrial.
The court heard that concerns about the credibility of the key prosecution witness were not disclosed to the defence.
Four senior police officers - including Northamptonshire's chief constable and deputy chief constable - are being investigated over misconduct claims relating to Staffordshire Police's investigation into the murder.
Judges were told on Thursday of an affair between two police officers involved in the case, which may have had a bearing on the evidence given by the witness.
The key witness, Simeon Taylor, said at the trial he had witnessed what happened, but has since retracted his evidence.
Mr Nunes, an amateur footballer on the books of Tottenham Hotspur, was shot five times in an execution-style killing in a country lane after cutting in on a gang's lucrative trade moving crack cocaine from Wolverhampton to Aberdeen.
Levi Walker, from Birmingham, Adam Joof, from Willenhall, West Midlands, Antonio Christie, from Great Bridge, West Midlands, Michael Osbourne and Owen Crooks, both from Wolverhampton, were found guilty of his murder at Leicester Crown Court in 2008.
They were given minimum tariffs of between 25 and 28 years.
Four men will be released soon. However, Walker will remain in jail as he is serving a prison term for another murder.
All of them apart from Mr Crooks watched the court proceedings via video-link from prison.
The Independent Police Complaints Commission (IPCC) confirmed in December that it was looking into allegations of misconduct relating to the force's investigation.
It followed the men lodging their appeal.
Northamptonshire Chief Constable Adrian Lee and Deputy Chief Constable Suzette Davenport are being investigated, along with West Midlands Assistant Chief Constable Marcus Beale.
They were with the Staffordshire force at the time of the murder inquiry.
Staffordshire's Assistant Chief Constable Jane Sawyers is also being investigated.
Mr Lee joined Northamptonshire Police in 2009. He is also the head of the Association of Chief Police Officers' ethics portfolio.
In a statement, Staffordshire Police said it noted the appeal court decision but would not comment further due to the on-going IPCC investigation.
"The force will now consider today's judgement - and the rationale for the Crown Prosecution Service decision - before determining the way forward," a spokesman said.
"We have assigned family liaison officers to offer support to Mr Nunes' family and keep them informed."
Speaking outside the court, Owen Crooks's mother, Jean, said the convictions being quashed was "fantastic news".
"It shows that justice has been done."
A major search is under way for a man believed to have fallen from cliffs at Arbroath in Angus.
[NEXT_CONCEPT]
Five men jailed for a gangland murder have been cleared on appeal.