The 26-year-old is believed to be ready to take the next step and join a club playing in Europe to help boost his England chances.
Bournemouth are not welcoming any interest in Solanke, who they have under contract until 2027.
It is only the presence of a £65m release clause that could force them to sell the former Liverpool forward, who scored 21 goals in 42 games last season.
Spurs boss Ange Postecoglou said last week that trying to sign a forward was a “focus”.
“It’s still the area of the park we’re really probably the thinnest,” he said.
Tottenham have interest in alternative options – including Lille’s Jonathan David – and are also open to signing a winger.
Richarlison faces an uncertain future amid reported interest from clubs in Saudi Arabia, and Spurs are keen to sign another striker whether or not the Brazilian leaves.
Forwards Bryan Gil and Alejo Veliz have already joined Girona and Espanyol respectively on season-long loan deals, though youngsters such as Dane Scarlett, Will Lankshear, Jamie Donley and Mikey Moore are all pushing for squad roles.
Spurs have signed midfielders Lucas Bergvall and Archie Gray and winger Yang Min-hyuk as well as re-signing striker Timo Werner on a season-long loan deal from RB Leipzig this summer.