Albion are keen to add some additional firepower to their Premier League strike force and Deeney's top flight experience could be a major factor in their decision.
Neal Maupay and Aaron Connolly are the main front runners for Brighton this season following Glenn Murray's loan switch to Watford.
They also signed the talented but inexperienced Andi Zeqiri from Swiss Super League side Lausanne-Sport. Zeqiri is seen as one for the future but Deeney is a proven striker and could make an instant impact.
A number of Premier League teams are in the running, scroll down and click through to see the latest odds, according to Sky Bet.
1. Brighton
Albion are 7/2 favourites to sign the Watford man
2. West Brom
The Baggies are priced at 5/1
3. Southampton
The Saints are 16/1 to sign the experienced striker
4. Aston Villa
The Villians are in contention at 33/1