The Spanish centre-back will miss the clashes with Arsenal, Manchester City and Crystal Palace after scans showed the foot injury that ruled him out of the 2-0 win over Stoke was worse than initially feared.
“A scan this week revealed a broken metatarsal,” said a statement on the official club website.
Amat, 23, has made 12 appearances this term and signed a new one-year contract extension in March to keep him at
Swansea until June 2018.
In Amat’s absence, Swans manager Garry Monk has full-backs Dwight Tiendalli and Jazz Richards back at Liberty Stadium following loan spells at Middlesbrough and Fulham respectively.
Watch Swansea in action away to Arsenal in the Premier League on Monday Night Football on Sky Sports 1 HD from 7pm.