The Chiliean international wasn't involved in Sunday's 2-0 defeat against Manchester United in the Community Shield with the club stating that Beausejour had suffered a thigh strain.
However this morning the rumours are that both Everton, managed by former Latic Roberto Martinez, and French Ligue 1 side Bordeaux are both interested in the 29 year old.
There has been no official comment from any of the parties involved, but the move would appear likely given the Spanish speaking exodus from the club following senior Martinez's departure.
Franco di Santo, Antolin Alcaraz, and Mauro Boselli are among the Spanish speakers who have already departed this summer as Coyle brings a distrinctlively British look to his side.