way.qub.ac.uk writes (single angle brackets):

ng 

There is no question of similarity in Jesus indication about John.
The passage in Matthew is very direct.  Where Luke (1:17) reports
the angel Gabriel prophesying that John will go before Christ "in the 
power and spirit of Elias", In Matthew 11: 14, Jesus himself says of John,
   
   "And if you care to accept it, he himself is Elijah, who
    was to come".

It is interesting that Jesus prepended the words, "If you care to accept
it", as if to say that the implications of this truth, namely of rein-
carnation, I will not force on you, but for those who can accept it, here it
is.  A Jewish poster to other newsgroups on Jewish esotericism and other
topics has outlined the esoteric, cabbalistic Jewish teaching of of
reincarnation and Karma, a teaching that is little known among Jews
today, but which is apparently widespread enough in Israel that Hannah
Hurnard ("Hinds Feet on High Places") was told about it by a Rabbi
she was trying to convert back in the 1940s as a missionary in Palestine.
Thus there may well have been a small number of Jews who knew about this,
whereas the large number of people did not.  The statement of Jesus about 
John, the greatest human personality in the New Testament, is guarded
but nevertheless quite direct.  Again, the subject of reincarnation, one
way or another, is not a subject of the New Testament, nor is the fate in general
of the human being between death and the last judgement.  But there are 
occasional indications that point to it.

As for the "popular belief" that Elijah would come again, it was more than 
a popular belief, as Jesus confirms it in more than one place, and he never 
corrected those who were expecting Elijah --  for example, those who thought 
that Jesus himself be he.