During the first half of the 2015-16 campaign, Eder failed to score for Swansea City in 15 appearances in all competitions, but he became a national hero in Portugal when he scored the winning goal in the Euro 2016 final against France.
However, despite featuring regularly for Lille since first moving to France in January 2016, the 29-year-old only netted six times during the most recent campaign and this has left the possibility of a transfer.
According to L'Equipe, both Brighton and Birmingham are keen to take advantage of his potential availability by lodging a season-long loan bid for a player who had impressed at Braga before making the switch to Swansea.
It has been suggested that Lille would prefer to cash in on Eder for a fee in the region of £5m, but the two English clubs may not want to shell out such a figure for a striker who failed to make an impact in his previous spell in the Premier League.
Brighton have made over half-a-dozen signings since gaining promotion to the top flight, but Chris Hughton is yet to add a central forward to his squad, while Birmingham boss Harry Redknapp is keen to add experience to his group of players ahead of a bid for promotion from the Championship.