Targett moved on loan to The Cottagers in January as they made a promotion push from nowhere, with the young left-back winning 16 of 21 games with the West London club.
It's clear that Targett made a huge impression on both the club's hierarchy and their fans last season, with the size of the opening bid reflecting this.
Will Saints sell?
Early reports suggested that Southampton would be reluctant to sell a player who has been labelled as 'one for the future'.
Yet, at 22, Targett needs to start showing the potential promised when he first broke through.
£10m isn't exactly pocket change either.
Combine that fee with the impending sale of Dusan Tadic to Ajax, funds will be needed to find an adequate replacement - and selling an infrequent third-choice left-back could be the best option, especially if the Saints can command a healthy fee.
Saints could also knock the offer back and ask for more money, in the hope of an improved offer or to ignite a bidding war between clubs - as Newcastle have been previously linked, with boss Rafa Benitez reportedly impressed with the youngster.
Why are Fulham interested?
Targett has many qualities which would improve Fulham as they embark on their first season back in the top flight of English football.
The former Scotland youth international is an exceptional crosser, notching crucial assists against Millwall and Reading last term.
Targett's wicked left foot can also pick open key passes, with the defender making 1.7 in an average Championship game last season.
The Southampton youth product also has many admirable defensive qualities, with interceptions and tackling some of Targett's strong suits.