No where really close to the city center. Teterboro is not worth the hassle, White Plains is a possibility - try Panorama, they have a fleet of new 172's, Morristown New Jersey has several schools and is a good bet. Further afield at Islip as mentioned there are several schools as well.
Going into Connecticut there are 2 or 3 nice schools at Danbury (about 60 miles North of NYC) and also at Bridgeport - 3 wings aviation amongst them. I can supply more details if you want.