Is 2. ..c6 a good idea in English opening?

The line with Nf6 first then c6 (in response to Bg2 or Nc3) is playable. This move order may transpose but I can't really vouch for it's strength.

Early ...c6 can serve two purposes, it can either be a move that hopes for a transposition to the Slav or in the reversed Sicilian it is essentially Alapin reversed. Both ideas are definitely playable. I play quite often 1. c4 e5 2. Nc3 Nf6 3. g3 c6 (known as Keres variation) and tend to get a good play with Black, but White can counter it with 4. d4. If White does not do that, Black has good chances of getting a strong center and limiting the range of the bishop on g2. I'm not so familiar with the Slav-like lines.