Honestly, I wouldn't trust syllables.com because it doesn't offer the actual, spoken alternatives. "caramel" it decrees is 3 syllables, "crayon," 2. Trust your ear.
I, like you, have always heard it as feel plus D, but have heard the Fi'- eld. And, I'm sure, given the way English works, there are are people you say "feel" as two syllables /fiːəl/.
That tells me it can be either. But then again, when it comes to language I'm a descriptivist rather than a prescriptivist.