Orthodox Islam does not accept reincarnation. Some Sufis (mystical Muslims), however, do accept it.
How can you deny God, when you were dead and God gave you life? Then God will cause you to die, and then revive you, and then you will be returned to God.
It is Allah that takes the souls (of men) at death; and those that die not (He takes) during their sleep: those on whom He has passed the decree of death, He keeps back (from returning to life), but the rest He sends (to their bodies) for a term appointed verily in this are Signs for those who reflect.
We have decreed Death to be your common lot, and We are not to be frustrated from changing your Forms and creating you (again) in (forms) that ye know not.
I died to the inorganic state and became endowed with growth,
And then I died to (vegetable) growth and attained to the animal.
I died from animality and became Adam (man):
Why, then, should I fear?
When have I become less by dying?
At the next remove I shall die to man, that I may soar and lift up my head amongst the angels;
And I must escape even from the state of the angels: everything is perishing except His face. (Mathnawi 3:3901-4, translated by R. A. Nicholson)
Druze and Alawi
A belief in reincarnation is atypical for Islam. There are, however, some Islamic sects that believe in reincarnation, including the Druze and Alawi who are most numerous in Lebanon, Syria, and Turkey. These minority groups hold a tenuous position among Muslims, in some measure due to their belief in reincarnation, and are often considered by their mainstream Sunni or Shi’a co-religionists as heterodox or even heretical. Druze and Alawi differ in several particulars regarding how they describe the workings of reincarnation. (Anne Bennett, “Reincarnation, Sect Unity,and Identity among the Druze,” Ethnology, Spring 2006)