The 29-year-old right-back, who has two caps for Holland, joined Swansea from Eredivisie club Twente in September 2012 and has made 27 appearances for Garry Monk's side.
He started his career with Utrecht and Feyenoord before moving to Twente in 2009, representing them in the Champions League. Although he was born in Suriname, Tiendalli has represented Holland from junior level and formerly played for their U-20 and U-21 teams.
With the loan window closing on Thursday March 26, Boro have now bolstered the squad with two new defenders, having completed the loan signing of Fulham centre-back Fernando Amorebieta earlier this week.
The deal is subject to International Clearance.