She said she had been "very reluctant" to let Baby P be looked after by Ms Godfrey, but she was bound by the 1989 Children Act to explore placement options with family and friends.
In a witness statement given to the police and seen by the programme, Ms Henry said: "My impression of Angela was that she believed the local authority were over-reacting and that the explanation for Baby P's injuries were those of his mother's, that they were caused by rough play and by his head-banging."
She also revealed Ms Godfrey asked for "a large sum of money, possibly £320 per week" to look after the child, and continually pressed for his return home.
Baby P suffered no injuries while in Ms Godfrey's care.
Police began investigating Baby P's case after he was put on the child protection register.
But with no new evidence to support a prosecution, services manager Clive Preece instructed that the boy should go back home.
However, Ms Henry said she had delayed this because police had expressed a similar opinion to her own, that the child should remain "out of the care of his mother".
In a statement, Haringey Social Services denies that Mr Preece overruled social workers.
"No concerns were raised regarding placement with Angela Godfrey at the time of the placement," it said.
The programme has also learned that police and social services had a serious disagreement over the care of Baby P after he was admitted to North Middlesex Hospital in June 2007 with more non-accidental injuries.
A confidential police report said a "frank exchange of views" had taken place.
Despite the disagreement, police eventually agreed to sign up to a care plan that meant Baby P was returned home once again.
In response to notes made in the police report, Haringey said rigorous discussion was expected between professionals when dealing with child protection, and that police had agreed to the final plans for Baby P.
The confidential police document goes on to criticise social workers for being too optimistic about the mother and not focused enough on the child.
Haringey said police did not express such views during discussions, and some had been formed "with the benefit of hindsight".
Programme makers also discovered that one hospital visit was missing from the police document.
Baby P was admitted to hospital in April 2007 for a large swelling to his head.
Sources told Panorama that police were never told about the hospital visit at the time, meaning the Metropolitan Police's child protection team was missing vital detail.
Haringey said it did not inform police because child protection issues were not flagged up by the hospital, and it had followed procedures.
Haringey Liberal Democrat councillor Neil Williams has called for social workers' and police officers' concerns to be taken more seriously.
"I think we need to move towards a system whereby if people are concerned then it escalates to the highest level immediately, and I'm talking about the chief executive of the council, the leader of the council," he told BBC Radio 4's Today programme.
Speaking in a separate BBC interview, Sue Berelowitz, deputy children's commissioner for England, said properly trained staff and good assessments were "critical" to ensure children were taken into care when necessary.
Ed Balls said he was "deeply disturbed" by failings of practice
"There is a received wisdom, and for very good reason, that it is much better to keep children in their families as much as possible really," she told Radio 4's Today programme.
Children's Secretary Ed Balls has asked Lord Laming to conduct a national review to see whether the recommendations the peer made in 2000 into the death Victoria Climbie - also in Haringey - have been put into practice.
The eight-year-old girl was murdered in Haringey in 2000 and the official inquiry into her death found she had been let down by several agencies.
Mr Balls confirmed Lord Laming's review would consider the independence of local safeguarding boards.
Panorama: What Happened to Baby P? will be broadcast on BBC One at 2030 GMT on Monday 17 November
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