Taylor broke and dislocated his ankle in a challenge with Sunderland's Craig Gardner, and was expected to miss the rest of the campaign. But the Wales left-back's rehabilitation progressed quicker than expected, and from the end of November Taylor had expressed his belief he could return before the summer.
Now Laudrup has said: "Physically he is quite close to being back. He has trained with us for some weeks, but we all know match fitness is a different level and only by coming back will he get that. The physical part is one thing, the mental part is another after a triple fracture, which is quite an injury. I am happy he's back and let's see when he can come in and make his comeback."
This weekend's trip to Norwich will come too soon for Taylor, but Laudrup has been boosted by the fitness of Chico Flores and Angel Rangel for the clash with the Canaries.
Flores aggravated an ankle injury having returned from seven weeks on the sidelines against Tottenham, while Rangel missed the 2-1 defeat with a calf problem, but both have returned to training.
Kemy Agustien is also closing in on a return from a thigh injury.
Laudrup said: "Chico and Angel have been training this week and should be back. Kemy has also been training with us for a couple of days."
Swansea head to Carrow Road on the back of three straight defeats and will be hoping to turn that run around as they look to hang on to a place in the top 10.
Norwich, on the other hand, are still in the midst of a battle against relegation, and Chris Hughton's side have won just one of their last 14 Premier League games.
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