As well as United and Spurs, Manchester City, Liverpool and Arsenal are also thought to be tracking the 19-year-old, who is reportedly set to quit his Championship club in January.
Interest in Zaha has skyrocketed in recent months, after a series of stunning performances in the Championship piqued interest in one of the most talented teenagers in British football.
The report suggests Arsenal are weighing up a bid for the youngster, while City and Liverpool have made tentative inquiries with a view to making a move when the transfer window reopens.
Steve Rowley, Arsenal's chief scout, watched Zaha at the King Power Stadium last weekend as Palace beat Leicester 2-1, and United and Spurs were also thought to have sent representatives to run the rule over the teenager.
Since making his debut aged 17, Zaha has gone on to make 107 appearances for the Championship club, and the 19-year-old - widely regarded as one of the most exciting wingers in England - is valued at £20m by Palace.
But the report suggests the Premier League clubs are more likely to open the bidding around the £10m mark. Liverpool, Manchester City and Arsenal have shown an interest for longer, but the arrival of Manchester United and Spurs could spark a bidding war in January, driving Zaha's price up further.
Liverpool and Spurs are thought to be eager to improve their wide options, while Manchester United, Arsenal and City and keen to bring the Football League's 2011-12 Young Player of the Year to the English top flight.
Crystal Palace has proven to be a hotbed of footballing talent over the past few seasons. In recent years Victor Moses, Nathaniel Clyne, and Zaha have come through the academy, and there are high hopes in south London for young Wales youth international Jonathan Williams.