The Spanish centre-back missed last weekend's 2-0 home win over Stoke with a foot injury and will now be absent for Swansea's final three league games against Arsenal, Manchester City and Crystal Palace.
"A scan this week revealed a broken metatarsal," said a statement on the official club website.
Amat, who signed a new one-year contract extension in March to keep him at Swansea until June 2018, has made 12 appearances in all competitions this season.
The 23-year-old provided left-back cover in Neil Taylor's recent absence and Swansea have recalled Dwight Tiendalli from his loan spell at Middlesbrough
Right-back Jazz Richards is also back at the Liberty Stadium following his loan move to Fulham
Source : PA
Source: PA