According to the Independent, Routledge and Leroy Lita will both undergo medicals at the club on Tuesday ahead of putting pen to paper.
A fee was agreed with Newcastle for the services of Routledge a while ago, but with serious interest from the likes of Nottingham Forest and big spenders Leicester, striking a deal has taken longer than expected.
But it now seems that Swansea are set to secure the signature of another one of Brendan Rodgers' top attacking transfer targets, to complement the other forward additions of Lita from Middlesbrough and Danny Graham from Watford.
"The bottom line is that we need to have a good squad, and these two are players who are going to help us stay in the league," Rodgers said at Swansea’s pre-season training camp in Austria last night.
"They have both played in the Premier League before, they are both hungry to push on and they both have pace."