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. He 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."
He added: "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."
If Bolton were to sign Williams it would be the second time in six months they had raided Boro for a defender following David Wheater's January switch.
The 24-year-old is behind Gary Cahill and Zat Knight in the central defensive pecking order at the Reebok Stadium but he may well retain his place as an emergency right-back for the visit of Sunderland on Saturday.
With Sam Ricketts a long-term absentee after rupturing his Achilles earlier this season and Gretar Steinsson still struggling with a knee problem, Coyle finds himself short of options.
The Scot remains hopeful Steinsson and striker Daniel Sturridge, who missed the defeats by Fulham and Blackburn with a hamstring injury, will be fit but midfielder Mark Davies (ankle) is unlikely to play.
Coyle said of Sturridge: "I'm hoping he'll be able to do some training today, and Gretar Steinsson as well. They haven't trained to this point.
"Sunderland might come too soon for Mark Davies but hopefully he'll be available for Blackpool and Manchester City, so bit by bit hopefully we can get the ones available to us back and have a healthier squad numbers wise than what we've had of late."
Source: PA
Source: PA