Bruce is keen to add a goalscorer this month to help build on his side’s impressive first half of the season.
Sunderland have priced their former manager out of a move for Steven Fletcher, while Shane Long now looks set to stay at West Brom.
Jelavic has been told he can leave Goodison Park this month, after losing his first team place to on loan Chelsea forward Romelu Lukaku.
London trio West Ham, Crystal Palace and QPR are all keen to sign the Croatian forward, but none have managed to match the asking price.
West Ham in particular are desperate to sign a new striker this month, but Sam Allardyce is reluctant to pay the £4.5million Everton are demanding, and is instead scouring the market for loan signings.
Bruce has £5million to spend on a new striker this month, and has also considered Filip Djordjevic of Nantes and Georgios Samaras of Celtic.
Everton want to sign a replacement before letting Jelavic go, and are currently weighing up various targets.