Mooney, 23, bagged 19 goals in 33 appearances last season but was unable to prevent Longford from being relegated.
There have been offers for the hitman from the Canaries, along with Crystal Palace and Colchester, but the player is now reported to favour a more conservative switch to remain in the Eircom league with either St Patrick's Athletic or Drogheda United looking favourites.
Cork City, who defeated Longford in last season's FAI Cup Final, are said to have dropped out of the race.