Saints have been in advanced talks for Dragowski, who spent last season on-loan with Empoli.
However, Sportitalia says the Cherries are now in contact.
Southampton have offered €10m, while Bournemouth have dived in with an improved proposal at €11m plus €4m in performance-related bonuses.
Roma are also believed to be tracking Dragowski's progress.
Need a VPN? Or seeking to switch? Then support Tribal Football by clicking here to sign up with our official partners NordVPN. Not only will you receive the best VPN experience, but also great discounts by signing up via Tribal Football. Thanks for your support!