I don't know if this can be verified, but I heard that more Muslims have converted to the Way of Jesus Christ in the last ten years than in all the other years since the advent of Islam in the 7th Century. I'm not sure it's an accurate figure, but I will say that something is certainly happening among Muslims and that there is an openness in their society that was not there before. I also want to point out that large numbers of nominal Christians, especially in Europe, are converting to Islam--a main reason being so they can marry Muslim women. Who has more converts? I have no idea. I will say that Muslims converting to Christianity often pay a heavy price in terms of persecution, and that Westerners converting to Islam are afforded generous protection by their governments.
But here is the question: why are Muslims attracted to the way of Jesus Christ? Here are some of the main reasons:
1) The Bible. We forget how compelling and convincing Jesus' teachings and parables are. He was certainly, among other things, a highly talented teacher, and his parables and sayings have the ability to lodge themselves into a person's mind, even after only one hearing. His insights into human nature, society, kindness, forgiveness, and God's power--not too mention the many miracles he worked--are convincing to a good number of Muslims. Of course Islam teaches that the Christian and Jewish Scriptures are muharraf--corrupted. But after a fair reading of the Gospels, sometimes the integrity and wisdom found therein can break through this Islamic doctrine. Incidentally, Muslim background believers from my experience tend to center their thought on the Gospels much more than Paul's epistles--very different than the evangelical tradition which has, in practice at least, tended to give primacy to the Pauline epistles.
2) Dreams, Miracles: What can I say? They are happening, and folks here don't have the initial inclination that Westerners do to "disprove" or "figure out how it was done." A miracle is from God--it's that simple. Often times in dreams people see Jesus, though at times it is one of the saints. This may not result in conversion, but it opens a path of inquiry that sometimes leads to conversion.
3) Charity: "They will know that we are Christians by our love." One refugee from a neighboring country received some help from some Christians, and he said, "We come here and we receive nothing form the Muslims--the Christians are the only ones who take care of us. I know nothing of your religion, but I will become a Christian." We think of the things like the Inquisition and the Crusades, but we forget the quiet, persistent witness of kind, caring Christians from all traditions, including Catholics and Orthodox. Which brings to mind something that Saint Francis, another apostle to the Muslims, said, "Let us not seek to be loved, so much as to love others." If you need to be reminded of the quiet work of the Church around you, just look at how many schools and hospitals are run by Christians or where founded by the various churches.
4) Christian Community: Islam teaches that a man should not speak to a woman unless they are married or of the same family. It is a witness to the beauty of the Christian community, and thus its faith, when Muslims see genuine respect and friendship between men and women who are neither related or married to each other. The assumption among Muslims is that this sort of thing must lead to fornication or adultery; on the other hand, young men and women sincerely yearn for fellowship with members of the opposite sex. When they see this among Christians it reveals that we are a peculiar people, that there is something different about us: fellowship, conversation, and friendship but without all the adultery and fornication they are told must result.
There are other reasons as well, but my experience is that these are the main ones. The challenge is getting people to think. Islam teaches that it alone is the reasonable and logical religion. Since people in the Middle East have extremely weak critical skills due to various reasons, this assertion is simply accepted. These are four things which I know have been important in challenging that assumption, which have led people to ask new questions and venture down new paths.
Peace be with all of you.