The Journal says Murphy will be the first player to leave Sunderland in the January transfer window with Cardiff leading the chase for the £1m-rated striker.
Murphy came in from the cold to make his first start of the season for the injury-ravaged Black Cats in Monday's draw at Blackburn and is likely to play again in the FA Cup third round tie with Barrow on Saturday.
Bruce has already sanctioned his exit "sooner rather than later".
Murphy is being tracked by up to six clubs, with Championship sides Queen's Park Rangers, Ipswich Town and Cardiff City at the head of the pack. Hull City - who were close to signing the Republic of Ireland forward in the summer - are also understood to be interested, while he is also among a number of back-up options being considered by Stoke City if they manage to sell the unsettled James Beattie during the transfer window.