The suggestions are that Burnley are ready to pay up to £400,000 for the 28-year-old who is in his second spell at Carrow Road.
Russell has one year left on his contract at Norwich after returning there two years ago following four years at Stoke. But with the Canaries having been relegated to League One it was evident that he would look to move on.
Preston have been favourites to land him, especially with the departure of Paul McKenna to Nottingham Forest, but it is believed the player is keen to move to Burnley.
With the squad and management out in America nothing is likely to happen until next week at the earliest.