Basically any opinion quoting a minimum amount has no backup... but a logical approach would say it needs at least 4 as else the hadith would be considered as a kind of ahaad (well strictly speaking even here the definition between earlier and later scholars differ so that the whole matter seems a bit spongy).
Imam as-Suyuti quoted 10 as a minimum.
Just to show that there's no real backup here some claimed minimum amounts and their "evidences":
Some said 12 based on:
and We delegated from among them twelve leaders. (5:12)
some said 20 based on the verse:
If there are among you twenty [who are] steadfast, they will overcome two hundred. (8:65)
70 as Moses/Musa() has chosen 70 of his people:
And Moses chose from his people seventy men for Our appointment. (7:155)
others said 313, the amount of the people whom participated at the battle of badr!
others made the claim 1700, the amount of the sahaba (known).
A-Shanqiti however states that there must be at least 4 witnesses, as is the minimum amount of witnesses for zina.
Imam ibn Hazm said tawatur can be by two witnesses, if we can exclude that they have lied!
Some sources so far (all in Arabic):
As-Sulaymani net quoting Shi'a opinion too.
Islamweb open link الحديث المتواتر to read details.
A thread on feqhweb.