Mr Linden, whose resignation as council leader led to the collapse of the SNP government in the region, resigned from the council earlier this year and left the party.
The Sunday Mail newspaper reported that five men had made allegations against the former leader, two of whom had spoken to investigators.
Mr Linden told the newspaper he did not accept the allegations made against him.
Continue reading: Visits from Humza Yousaf following allegations of sexual misconduct
A Scottish Police spokesman said: “Officers are investigating a report of a sexual assault in 2017. The investigation is at an early stage and is ongoing.”
On Saturday, SNP leader Humza Yousaf campaigned in Bellshill ahead of a by-election sparked by Mr Linden’s resignation from the council.
He said the party will investigate how it deals with complaints about Mr Linden.
Several councilors have also left the SNP group in North Lanarkshire to act as independents.
Mr Yousaf acknowledged that the SNP’s problems in this area “could have been handled better”.
He said: “We have a great local candidate here.
“We absolutely have our hands up – I say, as First Minister and leader of the party, things could of course have been handled better.
“That’s why we’re going to do the investigation.
“What we have now here in North Lanarkshire is a Labor Council which is frankly Conservative driven and that is not good for the people of this area at all.”