Mr Griffiths, 53, who has been the Labour MP for Edinburgh South for 22 years, faced allegations yesterday that he had had an affair with an unnamed woman including secret trysts within the Palace of Westminster, one of which took place on Remembrance Day last year.
The Labour MP, who is married and lives in Edinburgh, denied the allegations published in yesterday's News of the World.
Mr Griffiths's constituency of Edinburgh South is one of the most marginal in the country. At the last election, the Labour MP won with a majority of just 405 over the Liberal Democrats.
Not only do the Liberal Democrats think they can win the seat, but the Conservatives do too, and have made Edinburgh South one of their target seats in Scotland.
The allegations about Mr Griffiths led to some suggestions yesterday that his local Labour Party might decide to deselect him as the Labour candidate and replace him with someone less controversial in an attempt to hold on to the seat.
But a senior party source said the Edinburgh South constituency party was standing by Mr Griffiths and so was the Scottish party leadership. There was no consideration being given to deselection or any other moves to get another candidate for the seat.
It is understood that party leaders believe Mr Griffiths has a solid personal vote in the Edinburgh constituency and this would be lost if someone else was brought in to replace him.
Also, party leaders locally and nationally believe "the world has moved on" and voters are not as concerned about allegations of sexual impropriety by their public representatives as they were ten or 15 years ago.
Publicly, all Labour agreed to issue yesterday was a straightforward statement refusing to comment on the allegations.
A spokesman said: "This is a private matter, not a matter for the Labour Party."
Privately, Labour leaders said they "are not preparing to take any action" on the issue.
They do not intend to hold any meetings with Mr Griffiths to discuss the allegations or with the constituency party to find out what party members think.
The News of the World published pictures yesterday that, it claimed, showed an unnamed brunette stripping and posing in Mr Griffiths's Westminster office on 11 November last year before engaging in "sexual acts" with the MP. The paper claimed that the sex session was recorded on camera, as was another in a separate place, later that night.
The Scotsman made several attempts to speak to Mr Griffiths yesterday, but without success.
In the News of the World, the Labour MP described the claims as "absolutely groundless" and he argued that the newspaper had published "fabricated evidence".
In 2002, the Scottish MP was found guilty of breaking Commons rules on expenses and allowances, having claimed 10,000 a year in expenses for rent on an Edinburgh office that he owned outright.
The money was being paid into a trust fund he had set up for his autistic sister.
There were calls then for Mr Griffiths to resign or to be ousted from the Commons, but no further action was taken against him.
Mr Griffiths is a close ally of the Prime Minister – the two used to share a flat many years ago – and although Mr Griffiths has never been rewarded for his loyalty with high ministerial office, Gordon Brown has always stuck by the Edinburgh MP.