Police sources have told the BBC they knew of suggestions of a plot to kidnap Tony Blair's five-year-old son Leo.
Downing Street would not comment on alleged plans to kidnap Leo
The plan only got as far as what police called the "chattering stage", BBC correspondent Ben Ando said. There has been no kidnap attempt and no arrests.
The Sun said police believed a fringe section of campaign group Fathers 4 Justice discussed the idea but did not have the ability to do it.
Fathers 4 Justice has distanced itself from what it called a "grotesque" plan.
'Angry and upset'
Group founder Matt O'Connor said the news might force him to consider dismantling the group - a press conference will be held on Thursday to announce its future.
"It puts into question the whole existence of this organisation because I certainly wouldn't want to be associated with anything like this," he told BBC Radio 4's Today programme.
Mr O'Connor said he was aware that Scotland Yard anti-terrorist police had been investigating some former members of the group last month.
He said he believed Fathers 4 Justice had "contributed to the debate and raised awareness about the plight of children denied access to their fathers in the family courts".
He added: "I am very angry and upset that this organisation has been undermined by the very people it is supposed to serve."
Officially police are making no comment, and Downing Street has refused to confirm or deny the newspaper report.
The Sun claimed the plan was to hold Leo for a short period, as a "symbolic gesture" to highlight the cause of fathers denied access to their children.
Graham Dudman, the paper's managing editor, told the BBC: "The police took this very seriously because, of course, Fathers 4 Justice have been involved in some pretty spectacular stunts recently - breaching security at Buckingham Palace, at Downing Street, and also the throwing of the flour bomb at the prime minister in the House of Commons."
Security analyst Charles Shoebridge said Fathers 4 Justice had no record of "major criminality" and that this plot would have been "a very different matter - child abduction is itself a very serious offence".
"Let's not forget somebody could easily be killed in an attempt like this - not just the Fathers 4 Justice campaigners, if indeed they were Fathers 4 Justice, but an innocent passer-by could be hit in police crossfire," said the former Metropolitan Police anti-terrorism officer.
Ben Ando said police had been investigating extreme groups linked to Fathers 4 Justice for some time.
"The plan only got as far as what they [police] called the chattering stage," he said.
"No real reconnaissance was carried out, no actual kidnap attempt was made, no-one has been arrested, and the police are not convinced those at the centre of the alleged plan had the capability to carry it out."