The Daily Mail says City would like him to join Spanish international David Villa as a marquee player at the newly formed New York City FC, a prospect that appeals to the England midfielder.
And with the MLS season not starting until next Spring, City are also prepared to offer the 35-year-old a place in the squad at the recently renamed Melbourne City, that will play its A-League season throughout the British winter.
That would enable Lampard to keep playing beyond this summer before beginning a new career in America.
Negotiations with his representatives are not particularly far down the line at this stage.
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