The Muslim tradition reveres Jesus as a prophet but rejects his divinity. His sayings are found scattered throughout Arabic works on ethics, popular devotion in Islam, Sufi mysticism, collections of wisdom, and the histories of Muslim prophets and saints. The date for these sources ranges from 8th to 12th centuries.
Where do these sayings come from? Many of them echo the gospels, canonical and non-canonical, but not all. Most are deeply ascetic, as we know the Christian tradition was in eastern Syria which would have been the form of Christianity most well-known to Muslims. The Muslim Jesus speaks out to support Muslim definitions of piety. He is an advocate for religious responsibility.
It is important for us in the west to recognize that the Muslim Jesus and his sayings are the Jesus and his sayings that Muslims are familiar with - not the Christian Jesus. The prophets in Islam are men who set out to warn a proud or ignorant community, but whose messages are rejected. Yet God vindicates them in the form of some type of retribution. Jesus is a prophet involved in polemic: he wishes to cleanse his followers (i.e. the Christians) of their wrong beliefs. He denies tritheism (i.e. the trinity doctrine). His crucifixion is also denied, while his ascension becomes his vindication. He is a prophet with disciples around him. He is humble and pious, honoring his mother who is a virgin. His message is about God's unity or monotheism. There is no incarnation of a divine being, no crucifixion, and no redemptive death. His crucifixion is either portrayed as an event that only appeared to have happened, or someone like Simon the Cyrene or Judas Iscariot was made to look like Jesus and was crucified in his place.
This latter tradition has always been curious to me because it sounds like a received gnostic tradition such as has been attributed to Basilides by the heresiologists. The original gnostic teaching about this can be found in the Second Treatise of the Great Seth 55.19-20. What the teaching was is a development of Paul's idea that Jesus' crucifixion overcame the archons and powers (1 Cor 2:6-8; Col 2:8-15; Eph 6:12): Jesus could not be killed like other human beings because his spirit was incorruptible and without error. So "I did not die in reality but in appearance," Jesus says. His death actually happened to the archons and powers who nailed his body to the cross. Everything they did to Jesus, the archons and powers really did to themselves. They were tricked into defeating themselves by crucifying Jesus. Even Jesus didn't carry his own cross, the gnostics noted. Simon the Cyrene bore his cross in his place. I think that this tradition got mixed up (intentionally?) in the heresiologists' reports to credit the gnostics with saying that Simon and Jesus traded places, and Simon was even killed in Jesus' stead. This is the teaching that appears to have made its way into Islamic sources.