Linden, whose resignation as council leader led to the collapse of the SNP administration in the area, resigned from the council and left the party earlier this year.
The Sunday Mail newspaper reported that five men made allegations about the former leader, and two spoke to detectives.
Linden told the newspaper that she did not accept the accusations that had been made against her.
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A Police Scotland spokesperson said: “Officers are investigating a report of a sexual assault incident that took place in 2017. Investigations are at an early stage. scenery and ongoing.”
On Saturday, SNP leader Humza Yousaf was campaigning in Bellshill ahead of a by-election sparked by Mr Linden’s resignation from the council.
He said the party would investigate how he handled complaints about Linden.
Several councilors have also left the SNP group in North Lanarkshire to sit as independents.
Mr Yousaf admitted that the SNP’s problems in the area “could have been handled better”.
He said: “We have a great local candidate here.
“We are absolutely raising our hands. I, as prime minister and party leader, say that of course things could have been handled better.
“That is why we will do the investigation.
“What we have now here in North Lanarkshire is a Labor council supported, quite frankly, by the Conservatives, and that is not serving the people of this area well.”