The young winger is yet to play a minute in the Premier League since completing his £15 million move to Old Trafford in the summer.
Newcastle are believed to be leading the race for his signature, and Toon boss Alan Pardew is reported to have already put forward an offer for the Manchester United youngster.
However the Sun claim they face some serious competition.
Along with Newcastle, Swansea, West Brom, Aston Villa and Cardiff are all reportedly amongst those vying for his services.
Manchester United boss David Moyes has confirmed that Zaha is likely to go out on loan in January - although the Scot insists he has a long-term future with the reigning Premier League champions.
"I just don't think we have had the right opportunity to get him on as much as would have liked," he told Manchester United's official website last month. "We don't want to put him in too quickly. We want to make it the right time.
"I've said I would look at [a loan move] in January as I wanted to have him here for six months to see how we work.
I told him we would try to get him minutes but the competition in the wide areas we have with Nani, Antonio Valencia, Ashley Young, Adnan Januzaj and even Shinji Kagawa playing off the side means we are as strong there as anywhere.
Zaha, who helped Crystal Palace gain promotion to the Premier League last season, earned his first cap for England against Sweden last year.