As I understand it, a Catholic may hold a type of predestination to be true. I have pasted something from James Akin below. It is cut from his tiptoe thru tulip essay here.
The doctrine of unconditional election means God does not base his choice (election) of certain individuals on anything other than his own good will . God chooses whomever he pleases and passes over the rest. The ones God chooses will desire to come to him, will accept his offer of salvation, and will do so precisely because he has chosen them.
To show that God positively chooses, rather than merely foresees, those who will come to him, Calvinists cite passages such as Romans 9:15-18, which says, “[The Lord] says to Moses, ‘I will have mercy on whom I have mercy, and I will have compassion on whom I have compassion.’ So it depends not upon man’s will or exertion, but upon God’s mercy… So then he has mercy upon whomever he wills, and he hardens the heart of whomever he wills .”
What would a Catholic say about this? He certainly is free to disagree with the Calvinist interpretation, but he also is free to agree. All Thomists and even some Molinists (such as Robert Bellarmine and Francisco Suarez) taught unconditional election.
Thomas Aquinas wrote, “God wills to manifest his goodness in men: in respect to those whom he predestines, by means of his mercy, in sparing them; and in respect of others, whom he reprobates, by means of his justice, in punishing them. This is the reason why God elects some and rejects others… Yet why he chooses some for glory and reprobates others has no reason except the divine will. Hence Augustine says, ‘Why he draws one, and another he draws not, seek not to judge, if thou dost not wish to err.’” 
Although a Catholic may agree with unconditional election, he may not affirm “double-predestination,” a doctrine Calvinists often infer from it. This teaching claims that in addition to electing some people to salvation God also sends others to damnation.
The alternative to double-predestination is to say that while God predestines some people, he simply passes over the remainder. They will not come to God, but it is because of their inherent sin, not because God damns them. This is the doctrine of passive reprobation, which Aquinas taught .
The Council of Trent stated, “If anyone says that it is not in the power of man to make his ways evil, but that God produces the evil as well as the good works, not only by permission, but also properly and of himself, so that the betrayal of Judas is no less his own proper work than the vocation of Paul, let him be anathema… If anyone shall say that the grace of justification is attained by those only who are predestined unto life, but that all others, who are called, are called indeed, but do not receive grace, as if they are by divine power predestined to evil, let him be anathema.”