The newspaper reports the young parent exchanged letters and visited Watkins at his high-security jail.
He said he loved her and suggested she wear an engagement ring, it is claimed.
She said that during one visit he made a sick comment about her tot seeing her have sex.
Children's charity the NSPCC has said it is an "absolute shambles" that Watkins was allowed to carry on grooming from inside prison.
A spokesman told the newspaper: "We're sickened. It's utterly bewildering that he could carry on grooming.
"It shows contempt for children he abused, and raises serious questions about supervision."
According to a friend of the mother, she was under the impression Watkins was innocent and wrote to him to let him know people still cared about him.
The unemployed single mother then visited Watkins in November, and discovered he had put on four stone, gone grey and had rotten teeth.
She was shocked when he asked what she would do if her daughter "came into the room during sex".
She told a friend she felt "scared". But she insisted: "I won't let my daughter near him."
In December, police took away her mobile phone and laptop. Her child was taken from her care.
Staff from Wakefield prison told The Sun there was nothing untoward found in the pair's letters.
