The child, who came to be known as Baby Peter, was found on the doorstep of the vicarage of St Peter’s Church, in Hough Lane, Bramley, on November 23, 2016.
He was taken to hospital, where he was pronounced dead, but police yesterday said they had exhausted all lines of enquiry in their efforts to track down his mother.
The Rev Paul Crabb, who found Baby Peter, said in a message from everyone at the church: "Two years after the death of your baby, we recognise this is not over for you and not something you can simply forget as if it never happened.
“This baby still matters to you and you and he matter to us too.
“Please be assured we are always ready and waiting to help you in any way we can, maybe just by offering a listening ear and a decent cup of tea or coffee.
“Get in touch when you are ready.”
It was quickly established that the infant had reached full term and the placenta was still attached, so in all likelihood he had been left at the church soon after he was born.
It remains a mystery whether it was the mother herself or someone else who left the baby on the steps of the vicarage, which is not easily identifiable to those unfamiliar with the area.
The tragic events prompted an outpouring of tributes from the local community, with flowers, teddy bears and notes left outside the church as news of the baby’s death became more widely known.
Detectives now plan to speak to the Coroner's Office about finally releasing his body for a funeral.
Detective Superintendent Nicola Bryar, who stressed that the child's mother is not in trouble, has again urged her to get in touch to help with the arrangements for his funeral.
She said: "Our discussions with the Coroner’s Office will include when her son’s body can be released for a funeral. That is something we would like her to be able to lead the arrangements for and to have the chance to say goodbye properly.
“It is not too late for her to make herself known to me and I hope she might feel able to do that now.”