Real Madrid, Bayern Munich, Paris Saint-Germain, Arsenal and Liverpool have all been credited with monitoring the 19-year-old Brazilian.
Rayan, who only joined the Cherries from Vasco da Gama for an initial £24.7m in January, registered an assist on his debut and scored on his first start, form that has continued throughout his opening months on the south coast and earned him a first call-up to Brazil's senior squad ahead of this summer's World Cup.
Real Madrid have reportedly been building a detailed dossier on the winger, with new head coach José Mourinho, having seemingly secured Vinícius Júnior's long-term future, now turning his attention to squad depth and viewing Rayan's pace, close control and goal threat as an ideal fit for his attacking system.
Arsenal are also weighing up a move but must first resolve the futures of Gabriel Martinelli and Reiss Nelson to balance the books, while sources suggest the club believes it needs to act quickly given the scale of competition emerging for the teenager.
Liverpool, managed by Rayan's former Bournemouth boss Andoni Iraola, have discussed a fee in the region of £60m but see the Brazilian as a natural fit for their attacking system, even if Bradley Barcola remains their top target overall. Paris Saint-Germain's interest could intensify further still should Barcola leave the Parc des Princes, a scenario Rayan is understood to be following closely, with the French champions unlikely to baulk at matching Bournemouth's asking price given the size of fee Barcola himself would command if he departs.
Bayern Munich complete the list of suitors, though their interest remains at an early stage, with Bournemouth's current valuation considered too high by the German champions for now.
Bournemouth are under no pressure to sell, and with Eli Junior Kroupi facing up to four months out with a foot injury sustained in pre-season, the club will be especially reluctant to lose another of its most important attacking players this summer.
Even so, sources suggest the club's ownership would regard an offer in the region of £85m as exceptional value, leaving the door only narrowly ajar for any of Rayan's suitors willing to test that price.