The Magpies have had two bids - upwards of £40million - rejected for the 25-year-old after stepping up their interest over the weekend.
Time will tell if United submit a third offer with the Foxes believed to holding out for £60m, a valuation Eddie Howe’s side haven’t come close to meeting.
A successful move, according to Mills - formerly of Manchester City and Leeds United - would be a “step forward” and a “great move”.
Mills told talkSPORT: “It’s a step forward. You’ve got to look at where the two clubs are going.
“OK, so if you’re going to go off where they finished last year - that’s the only argument you have.
“Newcastle is a club on an upward trajectory. Leicester, at the moment, is on a downward trajectory.
“They’re about to lose Kasper Schmeichel, they’re losing players left, right and centre. They still haven’t brought a player in. (Youri) Tielemens could go, Wesley Fofana as well.
“Leicester, at the moment, looks like it is a club heading down. Their recruitment has been non-existent.
“Newcastle, with everything they’re putting in place and building for the future, would be a great move for him.
“He’s the sort of player Newcastle fans would love - a bit of creativity in that midfield.
“If I was him, I’d get myself on the A1 and start driving.”