The 24-year-old Holland international visited a specialist earlier this week after picking up the problem in the Europa League match against Metalist Kharkiv.
Manager Alan Pardew said: "His ankle is still a little bit swollen. We think he will be out for a maximum four weeks, five weeks, but we are hoping for less than that."
With veteran Steve Harper also injured, Rob Elliot was handed his Premier League debut in the 4-2 win over Southampton, and he will continue in goal at Swansea this weekend.
Fabricio Coloccini, though, is a major doubt for the trip to south Wales with a back injury he picked up against Southampton.
Pardew said: "I think he's doubtful but there's a chance he could play. He's someone now who we've got good options for if he's injured, and we didn't have that at all until Mapou (Yanga-Mbiwa) arrived.
"It is doubtful for Coloccini, I think, for Swansea and, if you'd taken him and Tim out of the team six weeks ago we wouldn't have a chance of winning this game.
"But, as it is now, we're a different proposition. We've got much more strength and of course we've got a chance of winning without them."