Brady joined the Canaries last summer from Hull City for a fee believed to be around the £7million mark and he has been a regular fixture in the first team ever since.
The 24-year-old played in all but two Premier League games last season but, despite being one of the stand-out performers, he was unable to prevent the club from being relegated to the Championship.
His impressive performances at Euro 2016 with the Republic of Ireland helped to raise his profile even more, with the defender scoring against Italy in the group stage and France in the last-16.
A move away from Carrow Road is highly expected and the left-back, who can also be deployed as a winger, could be set for a switch to Goodison Park.
Toffees boss Ronald Koeman is said to be a firm admirer of Brady and an offer around the £6-7m mark could be enough to land the Irishman in time for the start of the new season, despite Norwich reportedly wanting £20m