The 22-year-old, who has kept 13 clean sheets for the League One side this term, is regarded as one of the best shot-stoppers in the lower divisions.
Manchester United have been credited with an interest in the goalkeeper, while a number of Championship clubs are also reported to be monitoring proceedings.
"It is obviously flattering to be linked with clubs in the Premier League and Championship, but all my focus is on helping Brentford secure promotion," Moore told Sky Sports News.
"I am happy at Brentford and I want to keep playing regularly so I can continue my development.
Brentford currently sit third in the League One table.