English history pre-Conquest is complex and inextricably tangled with the Danes - in fact during this period the English were almost as much Anglo-Scandinavian as Anglo-Saxon. King Harold Godwinsson was half-Danish on his mother's side.
The Normans were in contact with England from an early date, given that they occupied most of the important ports opposite England across the Channel. However Norman influence in England really began in 1002 with the marriage of King Aethelred of England to Emma, sister of Duke Richard II of Normandy. (Reading this brief
biography of Emma will give you a reasonable idea of events during the pre-Conquest period). England at the time was plagued by Viking incursions, and Normandy (originally settled by Vikings) was often used as a Viking base for attacks on England. By marrying Emma, Aethelred hoped for a peaceful solution to that problem. After Aethelred's death, Emma married the new Danish king of England, Cnut, and acted as regent in England for a while after Cnut's death. Danish kings came and went until eventually Emma's son by Aethelred, Edward, was recalled from exile in Normandy and crowned king of England in 1043. Having spent much of his life in Normandy, Edward (the Confessor) was much more attuned to the Norman way of life than the English, and upset the English by bringing many Normans to England. He not only preferred them as associates, but also gave them many important offices, including earldoms and bishoprics.
The short answer is, though there were many Danish settlers in England, there wasn't a general wave of Normans entering England before the Conquest. Their presence only became noticeable during Edward's reign, when that relatively small group of Normans he favoured became an elite clique with an influential place in English government. Perhaps the hero in your WIP could be connected with Emma or Edward in some way? William the Conqueror’s claim to the English throne was through Emma as well - she was his great-aunt.