They disclosed that one woman told them that two youths had tried to entice her son away from the shopping centre shortly before James was abducted from the same centre by two boys on Friday.
Feelings have run so high over the murder of the two-year-old that at one stage up to 150 calls an hour from the public were received at the inquiry headquarters in Bootle. Traffic wardens and special constables were brought in to answer telephones.
Among those who have come forward are three girls who were on a security video which showed James being swung or dragged between the two boys on Hawthorne Road, half a mile from the shopping precinct. Detective Superintendent Albert Kirby, leading the inquiry, said: 'The information they have given us is absolutely tremendous.'
Police also regard evidence from a woman who was shopping in the T J Hughes department store in the shopping precinct at about 2.30pm on Friday as crucial. A police spokesman said: 'She turned round and saw two boys talking to her son. She went over and asked what they were doing, and they shrugged their shoulders and walked off.'
The mother, who has not been named, said: 'It is very frightening to think that what happened to James could have happened to my own child.'
The Independent has also learnt that two boys, believed to be those who abducted James, were begging in the precinct on Thursday and Friday. They approached a number of shoppers.
Several people have come forward from the Walton district, where James's body was found, with sightings of him with the two boys. Detectives are now certain that he was near Walton-on-the- Hill church on Friday evening. One woman saw the three by a florist's shop in County Road at about 5pm.
Police appealed for calm yesterday after angry crowds gathered outside the murder inquiry headquarters in Bootle and when a boy was arrested as a suspect on Tuesday night. They had to divert officers to deal with the disorder and made five arrests for public order offences. Chief Superintendent Dave Thompson, divisional police commander, said: 'It has probably put the inquiry back by at least 24 hours. We know there is a lot of frustration in the community and we appreciate that, and we are doing everything we can to achieve our main objective.'
The family of James Bulger joined the police in pleas for calm. Dennis Mather, brother-in-law of James's mother, Denise, said: 'I can only emphasise to the people of Bootle and the surrounding area to keep calm. The family are just as enraged as they are.'
He said members of the family had visited the inquiry headquarters. 'We didn't think there was enough being done. Now we have been shown round, we know enough is being done. The information coming in is phenomenal,' he said.
Ray Matthews, Mrs Bulger's brother, said that messages of comfort and support for the family had arrived from all over the world. People have sent cards, cheques and poems and they had received a letter from a mother whose child had been killed in a similar incident.
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