The 22-year-old Australia international is one of a number of players being monitored by the Trotters ahead of the reopening of the transfer window at the end of the season.
Coyle knows Williams, who can play at right-back and in midfield as well as in the centre of defence, from a loan spell he had at Burnley in 2009 during the Scotsman's time in charge at Turf Moor.
Coyle said: "Rhys Williams is a player I had on loan at Burnley when he was a young kid from Middlesbrough. He did terrific for me and he's one of 15 or 20 players that we've looked at.
"I spoke to (Boro chief executive) Keith Lamb yesterday because we have a fantastic relationship with Middlesbrough, and I spoke to Keith to say that's exactly where we are. If and when there was anything concrete on that then Middlesbrough would be the first to know."
Source: PA
Source: PA