The Dutch international has been in good form for the Swans this season, keeping six clean sheets from 11 home matches.
Such statistics have reportedly alerted the Premier League trio, although Swansea would be keen to keep hold of the 28-year-old.
"A number of clubs have approached us about Vorm," Van Zweden told Voetbal International.
"Manchester United have called, while Chelsea have made an approach as well. Tottenham Hotspur have also told us that they're interested. Vorm's departure would be a nightmare for us though."
Swansea paid £1.5m for Vorm during the summer.