As far as I can tell from the FARs as long as your training is performed by an FAA-certificated flight instructor and complies with the appropriate requirements of part 61 of the FARs you could theoretically do parts of your (primary/private) training anywhere you want, get all your endorsements, then come back to the USA to take your checkride and get a regular US (FAA) pilot's license for the appropriate category/class of aircraft.
From a practical standpoint I think this may be more work than the money you save.
If you do your pre-solo training in Mexico any instructor you come to in the US is going to want to fly with you a bit before they endorse you to solo.
If you try to do the whole thing in Mexico you would have to comply with Mexico's flight training requirements (Medical, student pilot certification, solo endorsements, etc. - about which I know basically nothing) as well the US FAA's requirements (like holding a US student pilot certificate with appropriate endorsements prior to taking your checkride).
Note that the procedure is a little different if you want to get a Mexican pilot's certificate and then convert it to a US/FAA certificate (that's all detailed in FAR 61.75 if you're interested. It doesn't sound like that's what you want to do, but at least as far as the regulations go that seems to be a more well-traveled path and something your local FSDO could probably advise you on - at the very least it would involve paperwork, possibly also a checkride).
I'd like to say a couple of words on Primacy since we're talking about training: Where you learn and the procedures you learn first will form the foundation of your aviation knowledge.
What that ultimately means for you depends on how different the regulations are where you do your training versus where you will do most of your flying. The FAA has been moving to standardize our procedures to be more in line with ICAO standards for a number of years now, but there are still differences and you may have to re-learn some of the regulatory bits of aviation if you did the bulk of your training elsewhere.
It makes sense to me to learn to fly where you intend to fly -- ideally in the same region (a sleepy corner of the midwest versus lower New York's airspace can be a culture shock!) but if cost is a factor find the best bang for your training buck in the airspace system you'll be spending most of your time in.