The 26-year-old began his career at Dutch giants Ajax, before going on to play for FC Utrecht and Feyenoord. He then signed for Twente but a three-year spell came to an end in the summer.
The former Netherlands Under-21 full-back joins the Premier League outfit on a year-long deal after Neil Taylor broke his ankle in the Swans' clash with Sunderland.
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The Welshman has since been ruled out for the remainder of the season, leaving manager Michael Laudrup in need of defensive options and thus prompting a move for the Dutchman.
Tiendalli completed his medical at the Liberty Stadium on Monday, with the necessary clearance expected to be granted before Swansea resume domestic action at the weekend.
The Swans have made a brilliant start to the new league season, winning two and drawing one of their opening three games, after losing manager Brendan Rodgers to Liverpool during the summer.
Those results mean that the Welsh side sit in second place of the Premier League, only bettered by Chelsea's perfect record, with a number of Laudrup's summer signings, in particular four-goal Michu, already shining.
The club's next fixture sees them travel to Aston Villa on September 15.