The 29-year-old is currently on loan at Queens Park Rangers, but his spell in London will be cut short to talk to Bristol City should a deal be agreed.
Sky Sports News reported earlier this week that Nottingham Forest and Bristol City have both made an approach to sign Wells.
Wells’ contract expires in the summer, although it does include an option to extend it for a further year.
The Bermuda international has scored 13 goals in 26 Championship appearances for QPR so far this season.
Burnley manager Sean Dyche confirmed on Friday that talks with QPR boss Mark Warburton are ongoing and the Clarets could recall the player with a view to selling him on.
Dyche said: “We have the ability to get him back here. I’ve spoken to Mark Warburton and let him know the different options there are.
“Over the last few days there has been a lot of interest in him, rightly so. He’s scoring goals in the Championship.
“He knows he’s well thought of here but he wanted and needed football. But we will make a decision on that, whichever way that one pans out.”
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