The donation - the largest in the party's history - was given by businessman Michael Brown through his company 5th Avenue Partners before the 2005 General Election campaign.
Brown skipped bail while awaiting a fraud trial last year and was sentenced to seven years in jail after being tried in his absence.
A spokesman for City of London Police confirmed that a complaint had been received about the donation.
"We have received a complaint against the Liberal Democrats. As is standard procedure, we are looking at it before deciding whether to launch a full investigation," he said.
The head of the economic crime unit, Detective Chief Superintendent Stephen Head, told the BBC: "On the face of it they seem to be serious allegations."
The Lib Dems have insisted the donation was received in good faith and was "properly spent" on its General Election campaign but some of Brown's victims claim the party used their stolen money.
Robert Mann, a tax lawyer in California, handed over five million dollars (£3.125 million) and is suing the Lib Dems for the return of £632,000 which Brown gave to the party.
The rest of his money was siphoned off for Brown's £2.5 million private jet.
"These monies belong to me," Mr Mann told the BBC's Newsnight programme last night. "The monies, as we found out, or a big part of them, wound up with the Liberal Democrats and, in my mind, anybody that did even a modicum of checking would have known that these monies belonged to me and not to Mr Brown.
"It's absolutely shocking how any major political party or any political party in England wouldn't want to return monies that are clearly stolen, absolutely traceable to me personally, and the individual that gave them the money is now a convicted felon."
A spokesman for the Liberal Democrats said: "All our donations from 5th Avenue Partners were received in good faith and were properly spent on the General Election campaign.
"Our auditors have seen our legal advice on this matter and are satisfied that we do not need to make provision for repayment."
In November last year, Brown, 42, was found guilty at London's Southwark Crown Court of stealing millions of pounds from ex-Manchester United boss Martin Edwards.
Brown posed as an international bond dealer, pretended his father was a Lord, claimed connections with royalty, and promised investors returns of up to 50 per cent.
His numerous lies helped him pocket nearly £8 million from Mr Edwards.
He was convicted of two thefts, one count of furnishing false information and one of perverting the course of justice between February 9, 2005 and April 17, 2006.
After the jury reached its verdicts, Martin Edmunds QC, prosecuting, told them Brown had previous convictions and had twice skipped bail.
Mr Edmunds said Brown was first arrested three years earlier over allegations of perjury and making a false passport declaration following concerns by HSBC over the management of clients' cash.
The perjury involved pretending entitlement to millions of pounds in assets frozen by the bank, while the passport offence concerned replacing the one he had been ordered to hand over to police.
Shortly afterwards, he fled to his "luxury hilltop villa in Majorca" but was found, extradited and, after admitting both charges, was jailed for two years.
The barrister told jurors that following his release he was allowed bail in connection with the matters they had tried.
He said "very strict conditions" were imposed, including a £250,000 surety from one of Brown's friends, but "in July it was discovered the bird had flown". He said information suggested Brown was abroad.
After the verdicts Lib Dem leader Nick Clegg told BBC News 24 "everyone recognised" the fact that his party accepted the £2.4 million donation "in good faith".
"It has been recognised that we did all the due diligence checks we could have done, totally unaware of the crimes of which Michael Brown has now been indicted," he said.
It was initially planned to postpone sentencing until Brown was found but last month Judge James Wadsworth QC told Southwark Crown Court that an international manhunt had proved fruitless and sentenced Brown to seven years in jail.
The judge gave him three six-year sentences for theft and furnishing false information, to run concurrently, and a further year for perverting the course of justice.
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