Both doubt and doubtful are often followed by clauses introduced by if, whether, and that. A choice among the three depends upon the kind of sentence involved. That is used when a negative or interrogative idea is involved: "There is little doubt that you are mistaken." "Can you any longer be doubtful that you are mistaken?" Whether is used in statements conveying genuine doubt and uncertainty: "It is doubtful whether he will live." "They doubt whether he was ever there." If is usually to be avoided after both doubt and doubtful, although some accomplished speakers feel that if and whether are interchangeable. Since the use of if is debatable in doubt constructions and the use of whether is limited, why not always use that?
Dictionary of problem words and expressions. Harry Shaw. 1975.