Add-ons could push the total package up to £25m, though sources close to the negotiations stress nothing has been finalised. The 21-year-old would sign a five-year contract at the Vitality Stadium if the move goes through, reports BBC Sport.
Real Madrid would be due 50% of any fee, having sold Rodriguez to fellow La Liga side Elche last summer. The forward is a Uruguay youth international but remains eligible to represent Spain at senior level.
Bournemouth's interest comes as they look to strengthen a squad that produced a record Premier League finish last season. That finish secured qualification for the Europa League.
The campaign was overseen by Andoni Iraola, who has since departed to take charge of Liverpool. Former RB Leipzig and Borussia Dortmund head coach Marco Rose has been appointed his successor on the south coast.
Bournemouth are targeting a striker and a left-sided centre-back this summer. Any forward brought in would add competition and support for Evanilson. Bournemouth are also willing to sell Turkey forward Enes Unal this summer, provided the right offer arrives.
The club are hoping for a smoother window than last year, when three of their first-choice back four and their number one goalkeeper all left the Vitality Stadium.
Elsewhere, Bournemouth are fielding interest in midfielder Alex Scott, who is valued at around £80m. Arsenal, Manchester City and Manchester United have all made enquiries.
Club sources, though, insist Scott is not for sale at any price this summer, regardless of the interest from three of their Premier League rivals. The coming days are likely to prove decisive in determining whether Rodriguez completes his move to the south coast.