The League One club are looking for a new boss following the recent sacking of Paul Groves.
Howe became the favourite to take over at Dean Court on Wednesday night.
"Bournemouth have made an unsolicited and unwanted approach for manager Eddie Howe," Burnley chief executive Lee Hoss told the club's official website.
"Burnley Football Club would like to stress that no agreement has been reached and Eddie Howe is still Burnley manager.
"No permission has been granted for Bournemouth to speak to Eddie."
Howe, who is under contract until June 2014, spent most of his 13-year playing career with Bournemouth before his retirement in 2007.