An original offer of £1.5m was rejected by United, but the Swans hope that a new bid of £1.8m will be successful.
Nottingham Forest are also keen on the former QPR man, but Swansea look to be in the driving seat for the 26 year old's signature.
Brendan Rodgers is a big admirer of Routledge and believes that he would fit into the club's style of play perfectly.
He said: "We've had a couple of bids that have gone in.
"We think he would fit into our style and excite our supporters. We will wait to see if we can agree a deal."