As senior LibDems refused to say how many claims of sexual harassment they had received since the accusations emerged, ComRes revealed the party had slumped to its lowest-ever rating in a monthly survey.
The poll gave the LibDems only 8% of the vote, putting it in fourth place behind the UK Independence Party.
Its rating is down two points on last month, compared to UKIP's 9%.
Crucially, the result comes three days before the Eastleigh by-election, caused by LibDem MP Chris Huhne's resignation for perverting the course of justice.
One of the women who had anonymously complained about Lord Rennard has revealed herself as Alison Goldsworthy, a party member from Wales.
Police also confirmed they had been asked to investigate "whether or not criminal activity has taken place".
The apparent escalation of the scandal came as pressure mounted on LibDem leaders to explain what they knew and when about the allegations.
LibDem Womens' Minister and East Dunbartonshire MP Jo Swinson would not be drawn on her earlier comments that she had taken "action" when concerns were raised with her.
But Lord Rennard did admit for the first time he had been confronted about rumours over his behaviour by now LibDem Chief Secretary to the Treasury Danny Alexander.
In an attempt to challenge cover-up allegations, senior LibDem figures also announced they had appointed an independent whistleblower for alleged victims to contact.
The party described Public Concern at Work as the "UK's leading whistle-blowing authority". Party sources also made clear evidence to the internal party inquiry into the allegations against Lord Rennard would be made public, as well as its results.
LibDem sources said the inquiry was expected to last three months and added that all the sexual harassment accusations against the former party chief executive, which stretch back to the early 2000s, would be looked at.
It will be headed by Alistair Webster QC, a former chairman of the Liberal Democrat Lawyers' Association.
Trials Mr Webster has been involved in include the Lady in Lake case, in which Gordon Park was convicted of murdering his wife 28 years ago and dumping her body at the bottom of Coniston Water, where it lay undiscovered until 1997.
Earlier LibDem president Tim Farron admitted his party "screwed up" its response to allegations of improper behaviour by Lord Rennard.
"There are individuals out there who we had a duty of care towards and we did not fulfil that duty of care," he said.
Last week the party insisted leader Nick Clegg had known nothing about the accusations before they emerged.
But on Sunday night the Deputy Prime Minister said his office had heard of anonymous non-specific rumours in 2008.
His then chief of staff Mr Alexander had challenged Lord Rennard, who denied the allegations. However, it has now emerged that Mr Clegg's chief of staff, Jonny Oates, had been given details in a 2010 email of the dates and locations of four alleged incidents between 2003 and 2007.
Last night Scottish LibDem leader Willie Rennie said the allegations had come as a surprise to him. It is also understood Scottish Secretary Michael Moore – a LibDem MP – also first heard of the accusations when they were made public last Thursday.
A Scotland Yard spokesman said: "The Metropolitan Police Special Investigations Command has been approached by officials in the Liberal Democrat Party and is working with them to ascertain whether or not criminal activity has taken place."
Lord Rennard has strenuously denied "any suggestion of improper touching" of women who he came into contact with in his role as chief executive.