The Blues won Europe's premier cup competition for the first time with victory over Bayern Munich on penalties at the Allianz Arena.
And Buck says that Abramovich, who has invested heavily in the Blues' squad since purchasing the club in 2003, now wants to build on that success.
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He told The Sun: "There’s a lot of clubs with three or four of those stars on their club badge. We have only one. Roman gave a thank you speech, the message was that it was all down to the boys."
Buck went on to discuss the future of Didier Drogba, whose late equaliser in Germany and final spot-kick proved crucial in lifting the trophy. The Ivorian is out of contract in just a month's time and is a reported target of Chinese club Shanghai Shenhua, now managed by former Blues striker Nicolas Anelka.
Buck continued: "We love Didier. But we have to think what’s right for Chelsea — and Didier has to do what’s right for him."
The American remained coy on the future of interim boss Roberto Di Matteo, however, but did admit that members of the squad are keen for the Italian to be appointed on a full-time basis.
He added: "We’ve heard from a number of the players that they’d like to see Robbie as manager.
"We haven’t set a timescale for a decision. Roman will be in London this week and we have to start thinking about it."