The Welsh club had already agreed an undisclosed fee with Newcastle and the 26-year-old will now undergo a medical tomorrow before signing a three-year contract.
Swansea confirmed the news on their official website this afternoon.
Routledge will be joined in south Wales tomorrow by Middlesbrough striker Leroy Lita, who will also undergo a medical after a deal worth £1.75million was reached with the Teesside club last week.
Should there be no complications, both players will come into contention for Wednesday's friendly against Celtic at the Liberty Stadium.
The pair continue manager Brendan Rodgers' summer recruitment drive following the arrivals of Danny Graham, Steven Caulker and Jose Moreira.