Bunley have had a fourth offer for Brighton midfielder Dale Stephens rejected.
The latest bid is believed to be around £5.5m.
A central midfielder remains a priority, after a summer-long pursuit of both Stephens, and Derby County’s Jeff Hendrick, for whom the club have had three offers rebuffed.
Stephens was an unused substitute for Brighton in their 2-0 defeat at Newcastle United on Saturday, and is out of contract in the summer.
But the Seagulls, who narrowly missed out on promotion to the Premier League in May, have shown a real determination to hold onto their star men this summer, knocking back Newcastle interest in winger Anthony Knockaert, while Crystal Palace have been offered no encouragement in their pursuit of centre back Lewis Dunk.
Burnley are expected to be busy in the build-up to Wednesday’s transfer deadline at 11 p.m.
On Saturday, Clarets boss Sean Dyche confirmed a deal for Chelsea striker Patrick Bamford is close to completion.
Burnley are set to take the 22-year-old forward on loan, to add to their striking options after the return of Rouwen Hennings on loan to Fortuna Dusseldorf, and Chris Long’s season-long switch to League 1 Fleetwood Town .
Bamford has yet to make an appearance for his parent club Chelsea, but was Championship player of the year while on loan with Middlesbrough in 2014/15.
Dyche said, asked whether he was taking the player: “It’s looking that way.
“It’s not quite done but it’s looking that way.”
As regards further incomings, he mused: “We’re certainly active. We do need support, and we’re in the market for support.
“I keep hearing all these stories, it would be nice if one of them comes true. I’d take one at the minute, it would be helpful if it was three or four.”
The summer window has been something of a struggle for a number of clubs so far, as he added: “The TV money can only be used with immediacy if you’ve got a backer who’s going to carry the can for the next three, four, five, six years.
“We’ve got wealthy people on the board but they’re not wealthy at that level.”