The Manchester Evening News says Zaha has been on Sir Alex Fergusonâ€™s radar ever since his sensational performance inspired Palace to a shock Carling Cup victory over United last November.
As a two-footed winger he is seen as a potential replacement for Nani if the Portuguese international is allowed to leave Old Trafford.
Ferguson is also considering Portoâ€™s Â£40m-rated James Rodriguez, but Zaha would represent a much cheaper alternative.
The England under-21 star could be available for around Â£15m.
That would also interest Roberto Mancini, with City already signing rising English talents Jack Rodwell and Scott Sinclair in the summer.