Norwich manager Paul Lambert turned down the chance to succeed Brian Laws at Turf Moor and Watford's Malky Mackay and Bournemouth's Eddie Howe have also been linked with the post.
Chairman Barry Kilby did not refer to specific targets but said in a statement: "Burnley Football Club is still in discussions with several parties with regard to the managerial vacancy at Turf Moor.
"Talks are ongoing and as a result I have informed caretaker manager Stuart Gray that he remains in full charge of first-team affairs for Saturday's Championship game against Queens Park Rangers at Turf Moor."
Laws left Burnley on December 29 following the club's dismal home defeat to Scunthorpe the day previously.
Source: PA
