See Jim Gatheral's Book "The Volatility Surface" (extract here http://janroman.dhis.org/finance/Volatility%20Models/lecture2%20Fitting%20vola%20skew.pdf)
where he obtains the squared implied volatility for strike $K$ as a time average of weighted expectation of square local volatility, then argues that the weight density is concentrated around a curve that connects the initial underlying price to the strike $K$, and finally approximates the density as if it was entirely concentrated on this curve, which he calls the most likely path approximation method. Also remember that the squared local volatility is the expectation of the squared instantaneous (stochastic) volatility conditional on the underlying price and you can view the squared implied vol as approximately the time average of squared instantaneous volatility along that most likely path.
Now if you use as your underlying the forward price and look at ATM implied vol, the most likely path can also be approximated with the constant path set to the forward price and the squared implied ATM vol becomes the time average of squared instantaneous vol along that constant path.
You might argue that the variance swap would give a true integrated expected square instantaneous volatility and thus be a better approximation in the context of computing the quanto adjustment for quanto options, but as I said in my answer to your original question Approximations for Quanto Options pricing there is so much uncertainty in estimating the FX / option underlying correlation that it does not really matter, the one thing you really want to make sure of is that your quanto adjustment is not strike dependent so that the quanto forward obtained by call/put parity is the same at all strikes.