The man tried to lure a schoolgirl into his car before following a young boy out riding his bike.
The 12-year-old girl, who was wearing her school uniform, was approached as she walked along Ludford Street at 1125 BST Friday.
The girl told police the man offered her a lift to school and then got out of the car and opened the passenger door.
When she refused he tried to grab her but she got away.
Shortly after 1600 BST the same day a man fitting the same description followed a 12-year-old boy riding his bicycle along Underwood Lane for up to four minutes.
It is thought he also followed a group of schoolgirls, and a boy on foot.
The suspicious man is described as white, 30 years old, about 5ft 9in tall with straight dark brown hair in a basin style.
Dirty jeans
Police said he wore a dirty, yellow bubble-style jacket and dirty black jeans.
His car was described as red Astra-type vehicle and possibly had white paint showing through at the back.
A Cheshire Constabulary spokeswoman said: "Police are warning parents to be extra vigilant at this time.
"We are asking anyone who may have any information regarding the identity of this man or information regarding these two incidents to contact police at Crewe."