The Norway international is expected to leave the Vitality Stadium in the coming weeks with only nine months left on his current deal, and the Cottagers are keen to offer him a chance to continue playing top-flight football.
Fulham aren’t the only ones interested in King, with Aston Villa also planning a move.
Manager Dean Smith wants another striker after landing Ollie Watkins, and his side have looked at a £16million deal but it could take more for Bournemouth to cash in.
Fulham have already brought in Antonee Robinson, Mario Lemina, Harrison Reed, Alphonse Areola and Kenny Tete so far this summer, and Ola Aina is expected to join from Torino.
They could do with another striker, and King will be a fine addition.
King was linked with a return to Manchester United in January as boss Ole Gunnar Solskjaer needed a cover for the injured Marcus Rashford, but Bournemouth knocked back their offer.
The 28-year-old has scored 48 Premier League for the Cherries since joining in 2015, hitting double digits in 2018-19 and 2016-17, and Fulham need such an experienced striker in their attack.