Arminianism (and most free-will theologies) hold that natural man will only choose to reject God because they are "flesh". But the Spirit works within man in such a way as to influence their desires (by the Spirit men can choose God).
I don't really have much against true Arminian free will but let me raise one question. If you really believe that natural man will only choose to reject God then you do believe in total inability and are an almost Calvinist, at the least. In other words, if the "Spirit works within man" in an essential manner, even if it is influence or conviction, if you admit that it is true that without it coming to Christ was not going to happen - then in practice there is not much difference between that and the Calvinist idea that one is born again and that results in a person coming to Christ. For in both cases you have a natural man lost, with no ability to respond on his own.
And then the hedging begins. What the Arminian objects to is God being sovereign in this to the extent that the only real difference between the one saved and the other lost is God's own choice in bestowing grace no matter the type of grace, whether conviction and enlightenment or being born again. They get around this by saying a certain amount of grace is bestowed universally as "prevenient grace" and it is of an enabling and persuasive nature - with man being able to reject it. But if that is the case, to me once you say it is universally bestowed you are back to all men then being given "enough" grace and the choice is then up to them, which to me is no different than a modern free willer who recognizes this as being a needless concession to Calvinism (from a time when it meant something to be called a heretic by Calvinists) and thus they go ahead and say what all free willers really believe - it is truly up to you, and you have all you need to properly decide for Christ and are responsible if you don't.
And please don't think I am being dismissive of Arminianism. I admit that I think there indeed does have to be at some level, whether God supplies convicting and enabling grace or if it's a regenerating grace - if there is not at some level, an ability to thwart this to the extent of losing the opportunity to be saved, then I cannot satisfy myself that this does not lead to a real possibility of God being blamed - as he is the only cause.
I know that a good Calvinist will say "who are you to reply against God", which is true enough if that was all we had. But from scripture, and from Puritan Calvinist preaching even, I don't think we have much evidence that God is doing it like that. There just is too much evidence that men are consciously and with full understanding either choosing unbelief, or preferring the advantages of the world and pleasures of sin more than Christ- and are truly making a free will decision to do so. I would welcome discussion on this because I do not have a settled position myself.