Spicer says Trump "continues to maintain that belief". There has been no evidence to support the claims that there was widespread voter fraud in the election.
Spicer's only attempt to support Trump's assertion was to point to a 2008 Pew Research survey that showed a need to update voter registration systems.
Donald Trump under fire as lawsuit filed over his business interests
Donald Trump's extensive business interests have made him rich and famous but could they also be his undoing?
Trump has barely had a chance to settle into the Oval Office but the billionaire TV star turned US President is already drawing fresh legal attacks from critics.
The American Civil Liberties Union tweeted within minutes of Trump taking the oath of office on Friday that it had filed a request under the Freedom of Information Act asking that government agencies hand over memos, emails and other private communications with Trump's transition team on his conflicts of interest.
Trump is also facing a lawsuit, petitions and a challenge from WikiLeaks over his refusal to sell his businesses or release his tax returns. He's even facing action over royalties from his TV show The Apprentice.
His company, the Trump Organisation, has stakes in golf resorts, office buildings, residential towers and hotel licensing deals in about 20 countries. Those include ones with which the US has sensitive relations, such as the Philippines, Indonesia, South Korea and Turkey.
With so many business ties, particularly abroad, government ethics experts worry US interests could take a back seat to his personal financial concerns.
And even if they don't, they argue, people will try to curry favour with the new President by buying apartments in his towers or memberships in his golf resorts, raising doubts - fair or not - that US policy is for sale.
But Trump said at his news conference earlier this month that he would not sell his ownership in his company, instead handing over management control to his two adult sons.
He pledged that his company would strike no more deals abroad and would donate any profits from foreign governments using his hotels to the US Treasury.
'CREEPING, INSIDIOUS THREAT'
You'll probably be hearing a lot more about the "Emoluments Clause" after a watchdog group on Monday filed a lawsuit alleging the US President is violating the Constitution.
The clause says that "No title of nobility shall be granted by the US: And no Person holding any Office of Profit or Trust under them, shall, without the consent of Congress, accept of any present, emolument, office, or title, of any kind whatever, from any King, Prince, or foreign State".
Scholars say the words were added because of concerns that American ambassadors might be corrupted by gifts from rich European powers.
Citizens for Responsibility and Ethics in Washington believe the clause prohibits Trump from receiving money from diplomats for stays at his hotels or foreign governments for leases of office space in his buildings.
It will seek a court order forbidding Trump from accepting such payments, said Deepak Gupta, one of the lawyers working on the case.
Trump does business with countries like China, India, Indonesia and the Philippines, the group noted in a statement.
"When Trump the President sits down to negotiate trade deals with these countries, the American people will have no way of knowing whether he will also be thinking about the profits of Trump the businessman," it said.
When Trump the president sits down to negotiate trade deals with these countries, the American people will have no way of knowing whether he will also be thinking about the profits of Trump the businessman.
The group believes it is fighting a "grave threat" to the country but the language in the clause is disputed by legal experts, and some think the lawsuit will fail.
Trump called the lawsuit "without merit, totally without merit" after he signed some of his first executive actions in the Oval Office on Monday.
Trump's son Eric Trump, an executive vice-president of the Trump Organisation, told the New York Times the lawsuit was "purely harassment for political gain".
He said the company had taken more steps than required by law to avoid any possible legal exposure, such as agreeing to donate any profits collected at Trump-owned hotels that come from foreign government guests to the US Treasury.
But the lawsuit signals the start of a legal assault on what Trump critics see as unprecedented conflicts between his business and the presidency.
The group is being represented in part by two former White House chief ethics lawyers: Norman Eisen, who advised Barack Obama, and Richard Painter, who worked under George W Bush. The two have expressed frustration that Trump has refused to take their recommendation and divest from his business, and feel they had no choice but to take legal action.
"As the Framers were aware, private financial interests can subtly sway even the most virtuous leaders," the lawsuit argues, "and entanglements between American officials and foreign powers could pose a creeping, insidious threat to the Republic."
At a news conference earlier this month, Trump Organisation lawyer Sheri Dillon said the so-called emoluments clause of the Constitution isn't meant to ban fair-value exchanges.
They didn't think "paying your hotel bill was an emolument", she said.
'MORE GRATUITOUS THAN CLINTON'
A public petition was also filed to the White House, the first one for the new Administration, calling for Trump to immediately release his tax documents so the public would know if the new President was violating the emoluments clause.
A public petition to the White House on Friday demanding he go public with his tax returns gathered more than 250,000 signatures - well over the 100,000 needed to trigger an official response.
"The White House response is that he's not going to release his tax returns," senior White House adviser Kellyanne Conway said on ABC's This Week. "We litigated this all through the election. People didn't care."
But some people do seem to care and WikiLeaks is also pushing for their release.
On Monday it put out a call for people to help it publish Trump's tax returns, tweeting: "Trump's breach of promise over the release of his tax returns is even more gratuitous than Clinton concealing her Goldman Sachs transcripts".
YOU'RE FIRED!
The lawsuit on Monday says that Trump's company is receiving payments from foreign government-owned tenants at Trump Tower in New York, including The Abu Dhabi Tourism & Culture Authority and the Industrial and Commercial Bank of China.
The lawsuit also argues Trump's The Apprentice could get him in trouble. It says government-owned stations in the United Kingdom and Vietnam are paying for broadcast rights of versions of the reality TV hit.
Eisen and Painter have urged Trump to sell his holdings and put the cash in a blind trust, following the example of recent presidents.
The group behind Monday's lawsuit also filed a complaint last week addressed to the General Services Administration, an agency that oversees the lease of the government-owned building that houses Trump's new Washington hotel.
The complaint argued the agency must cancel the lease because it expressly forbids any elected official from benefiting from it.
GSA officials had said they needed to wait until Trump took office before weighing in on the issue. They have yet to issue an opinion, though, and have not responded to repeated requests for comment.
In the new lawsuit, the group faces several legal hurdles, including making the case that it even has standing to bring the suit.
"There are a lot of issues that have to be litigated for the first time," said Noah Bookbinder, executive director of Citizens for Responsibility. He added, though, that "we have never had a President who has in a significant way accepted foreign payments."
-additional reporting AP