Senesi sees his contract at the Vitality Stadium expire next month, and several Premier League clubs are queuing up to land him on a free transfer.
Liverpool were believed to be showing an interest in Senesi, alongside Tottenham Hotspur, Chelsea and Manchester United, but it appears the Reds have ended their pursuit.
According to the Liverpool Echo's Paul Gorst, the Reds are more focused on sealing Ibrahima Konate's long-term contract at Anfield rather than landing the Argentine.
Gorst claims that with Virgil van Dijk potentially being joined by Konate, Giovanni Leoni and £60m signing Jeremy Jacquet next season, Liverpool are 'well set in the centre-back department'.
News of Arne Slot ending his interest in Senesi - who has starred in Bournemouth's charge towards Europe - will be music to the ears of other clubs sniffing around.
The 28-year-old has appeared in all but one of the Cherries Premier League matches this term, only missing December's 1-0 defeat to Everton through suspension.