There are some exceptions from the MM. PG 63 Quasit, has a familiar Variant. Imp has a variant familiar, pg 69, as well as Psedodragons, pg 254. Each of these would either have to be summoned specifically or encountered and have the bargain struck at that time, but once the deal had been sealed, they can be bonded in a similar manner as the familiars from the Find Familiar spell.
I have included the variant quotes as follows.
VARIANT: QUASIT FAMILIAR
Mortal spellcasters interested in extraplanar familiars find quasits easy to summon and eager to serve. The quasit plays the part of the obsequious servant. It serves its master well, but it goads the mortal to greater and greater acts of chaos and evil. Such quasits have the following trait.
Familiar. The quasit can serve another creature as a familiar, forming a telepathic bond with its willing master. While the two are bonded, the master can sense what the quasit senses as long as they are within 1 mile of each other. While the quasit is within 10 feet of its master, the master shares the quasit's Magic Resistance trait. At any time and for any reason, the quasit can end its service as a familiar, ending the telepathic bond.
VARIANT: IMP FAMILIAR
Imps can be found in the service to mortal spellcasters, acting
as advisors, spies, and familiars. An imp urges its master to
acts of evil, knowing the mortal's soul is a prize the imp might
ultimately claim. Imps display an unusual loyalty to their
masters, and an imp can be quite dangerous if its master is
threatened. Some such imps have the following trait.
Familiar. The imp can enter into a contract to serve
another creature as a familiar, forming a telepathic bond with
its willing master. While the two are bonded, the master can
sense what the imp senses as long as they are within l mile
of each other. While the imp is within 10 feet of its master,
the master shares the imp's Magic Resistance trait. If its
master violates the terms of the contract, the imp can end its
service as a familiar, ending the telepathic bond.
VARIANT: PSEUDODRAGON FAMILIAR
Some pseudodragons are willing to serve spellcasters as a
familiar. Such pseudodragons have the following trait.
Familiar. The pseudodragon can serve another creature as
a familiar, forming a magic, telepathic bond with that willing
companion. While the two are bonded, the companion
can sense what the pseudodragon senses as long as they
are within 1 mile of each other. While the pseudodragon is
within 10 feet of its companion, the companion shares the
pseudodragon's Magic Resistance trait. At any time and
for any reason, the pseudodragon can end its service as a
familiar, ending the telepathic bond.
Here is Jememy Crawford's stance:
Warlocks have a special feature that allows them to have a pseudodragon familiar. If anyone else wants such a familiar, you need to use the variant rule in the Monster Manual. #DnD