Arsenal were thought to be in pole position for his signature, but reports have emerged that their interest has cooled, opening the door for Spurs and United.
It is believed that both clubs are willing to allow Zaha to remain at Selhurt Park for the remainder of the season, with the duo keen for him to continue to develop.
It is likely that the Eagles will demand a fee in the region of £15m, but may accept a lower figure if they are offered a fringe player on loan for the rest of the season.
Zaha has enjoyed a fine start to the season, helping Palace in their bid for promotion to the top-tier of English football.
Despite his young age, and the fact he is playing outside of the Premier League, England manager Roy Hodgson called up the winger for his side’s friendly against Sweden in November.
This was speculated to be a measure taken by the FA to prevent the player electing to represent the Ivory Coast, his country of birth, later in his career.