Basham, who can play in midfield or defence, was the subject of a bid from Blackpool boss Ian Holloway on Friday, which Wanderers rejected.
Coyle would ideally like to retain the 22-year-old, but acknowledges it might be in Basham's best interests to move on in order to improve his chances of playing first-team football.
"Olly (Holloway) rang me on Friday and informed me he was interested," Coyle told the Bolton News.
"I said I didn't want to lose him because he's a young player that we've got high hopes for at this football club.
"But I can totally accept that young Bash wants to play.
"He probably feels the competition is fierce in the midfield and he maybe thinks he's down the pecking order a little bit.
"What I said was that if Blackpool put forward a bid that we thought was fair, then we would allow permission for them to speak to him, on the basis on being fair to the kid."
Coyle revealed several clubs had contacted him about both Basham and young forward Danny Ward.
"I've had numerous enquiries about Bash, whether it's to loan or to buy outright, the same as I have for Danny Ward - they are two young, talented players," Coyle said.
"I'll always look to do the best for the football club, do the best business possible, but at this moment in time Blackpool's offer is not acceptable.
"As a manager you have to do right by your players as well, and I'm always very transparent, and I have kept Chris in touch every step of the way."
Holloway hopes to make a series of signings over the next few days and is understood to be in negotiations to secure a deal for striker Marlon Harewood, who has been on trial with Blackpool following his release by Aston Villa.
Meanwhile, Bolton are expected to announce the capture of teenage frontman Tom Eaves from Oldham this week.
The 18-year-old scored a hat-trick against Wanderers in a pre-season friendly last month.