Coyle is keen to strengthen his defensive ranks this summer, and is understood to be ready to offer Williams around ?25,000-a-week to swap the Riverside for the Reebok Stadium.
Boro boss Tony Mowbray is desperate to keep the Australia international, who has two more years of his current contract to run.
But with chairman Steve Gibson keen to slash his club's ?20m wage bill at least in half, and with Julio Arca, Andrew Taylor and Willo Flood the only players due to become free agents this summer, Mowbray could be forced to cash in if he does not receive offers for the members of his squad he would prefer to move on.
Coyle is looking to strengthen his defensive options in the close season, and having already prised David Wheater from Teesside in January, the Trotters boss, who is currently having to play Wheater as an out-of-position right-back, is ready to make another swoop on Rockliffe Park.
He already has experience of working with Williams, as the 22-year-old spent five months on loan at Burnley when Coyle was the manager at Turf Moor.
Williams spent most of that spell playing at right-back, but the Boro Academy product is equally at home at centre-half or in central midfield, as evidenced by his displays since returning from injury in March.
Having missed the first seven months of the season because of a pelvic problem that forced his withdrawal from Australia's squad for last year's World Cup, the youngster has caught the eye in his eight starts and two substitute appearances under Mowbray.
His versatility is a major asset, and his comfort and poise on the ball suggests he is capable of handling life in the top-flight.
Coyle's scouts have been monitoring his progress closely, and the Bolton boss is ready to make a formal approach once the transfer window reopens at the start of June.
Mowbray is expected to rebuff his initial offer, but the longer it takes to move on other members of the squad, the more likely it is that a prize asset such as Williams will have to be sacrificed to reduce the wage bill and raise funds for squad strengthening.
A balancing act is required, as the last two months have suggested the current Boro squad is more than capable of challenging for promotion next season.
Monday's victory over promotion hopefuls Cardiff made it six wins, four draws and just one defeat in the last 11 matches, a run that would have put Boro in the promotion shake-up had it been replicated over the whole of the campaign.
We've got momentum going into the summer, but we need to try to keep as many players as we can, said Barry Robson, who skippered the side in the absence of the injured Matthew Bates on Monday evening. We've got some good players here, and hopefully they'll stay. Hopefully, we can hold on to most of our players and maybe add a few as well.
The core of this squad is more than good enough to be challenging for promotion next season. You only have to look at the pedigree we have in that dressing room and the level the players have played at to see that.
Monday's display certainly suggested that Boro are more than capable of competing with the best the Championship has to offer.
Three goals in the opening 21 minutes ensured Cardiff would not be celebrating automatic promotion, unless QPR are docked a significant number of points by this week's FA disciplinary hearing, but Robson did not feel Boro's dominance was a major surprise.
Instead, having watched the Teessiders score three goals at Ipswich and four goals at Hull in their two previous away games, the Scotland international was expecting another strong showing.
We've done that a few times over the last few months, he said. It shows you what good players we have at the football club, and I think it also shows you the false position we're in.
I've always believed we were a better team than our league position was suggesting. We've started to show that in the last couple of months and, on current form, we'd be right up there if we'd started the season better.
Source: Northern Echo
Source: Northern Echo