In the women's, Switzerland and Ukraine have shown glimpses in recent years but will be out to make big statements.
While Serbia might just feel inspired by the continual Ws from their men's counterparts as the No.8 seeds look to create their own history.
Men Pool A: Belgium, Estonia, Croatia Men Pool B : Germany, Italy, Ukraine Men Pool C: Switzerland, Romania, Denmark Men Pool D: Spain, Slovenia
Women Pool A: Spain, Switzerland, Ukraine, Serbia, Denmark Women Pool B: Romania, Italy, Estonia, Slovenia, Croatia
In the men's, No.1 seeds France will be hard to stop with big fella Lucas Dussoulier set to make the paint his personal playground.
But No.2 seeds Austria may have claims to being the top contender as they bring back the same four players who reached the quarter-finals of the World Cup on home soil.
Poland also made the quarters in Vienna and are itching to get back into the Europe Cup, where they claimed bronze in 2021 after Przemyslaw Zamojski turned into a flame-thrower.
If Zamojski heats up Bratislava then Poland will likely be booking a ticket to Jerusalem.
The women's draw is loaded with Poland having top billing as deadly duo Aldona Morawiec and Klaudia Sosnowska aim to shrug off a disappointing winless World Cup for the Poles.
Austria deserve to be rated among the contenders having enjoyed the home comforts to reach the quarters at the World Cup. Sigrid Koizar was in eye-catching form in Vienna and will arrive in Bratislava with plenty of swagger.
Hungary and the Czech Republic, who made the play-in for the World Cup, have been consistent performers over the years and also loom as genuine threats for the coveted tickets.
The men's draw is deep with several teams capable of upsetting the top seeds. Hungary is a prime candidate led by high-flyer Attila Demeter, who can spark his team with a game-changing dunk.
Strong hoops nation Greece will be looking to make a statement on the half-court, while Slovakia should enjoy the comforts of the home surroundings.
In the women's, hosts Slovakia will be banking on the support of the home crowd to fuel them to a giant-killing run while Portugal might just be a dark-horse with Joana Soeiro capable of winning games almost single-handedly.
Men Pool A: France, Turkey, Bosnia and Herzegovina Men Pool B : Austria, Hungry, Republic of North Macedonia Men Pool C: Poland, Slovakia, Portugal Men Pool D: Czech Republic, Greece, Georgia
Women Pool A: Poland, Austria, Greece, Slovakia, Portugal Women Pool B: Hungary, Czech Republic, Latvia, Turkey, Georgia
What's at stake in Constanta and Bratislava
In the men's, the winners of the semi-finals will qualify to the Europe Cup. The two semi-finals losers will play one additional game and the winners of that game will also qualify.
In the women's, the pool winners will qualify to the Europe Cup. The second-placed teams of each pool will play one more game with the winner to make it through.
A third qualifier will be played in Limassol, Cyprus, on June 17-18 to determine the final spots.