Wilson has been at St James' Park since joining from Bournemouth in September 2020.
The 32-year-old was very much Newcastle's main man up front initially but his role has been reduced slightly following the arrival of Alexander Isak in August 2022.
Wilson has not yet played for the Magpies this season after suffering a back injury during the summer.
He is expected to return to action next month, but his long-term future may lie away from Tyneside.
Striker Callum Wilson, 32, has scored 48 goals in 108 appearances for Newcastle United
According to The Sun, Fulham and Brentford are both monitoring Wilson, who's current Newcastle contract is due to expire in June 2025.
Brentford sold star striker Ivan Toney to Saudi side Al-Ahli on August 30.
Toney has since scored three goals in four games for the Jeddah-based outfit - who, like Newcastle, are owned by Saudi Arabia's Public Investment Fund.
Meanwhile, Fulham are also understood to be keen to bolster their attacking options, which currently include Raul Jimenez, Jay Stansfield and Rodrigo Muniz.
Jimenez scored the opener as Fulham beat Newcastle 3-1 on Saturday.
Brentford boss Thomas Frank, pictured on Saturday, is on the lookout for a new central striker
Newcastle boss Eddie Howe provided an injury update on Wilson ahead of that fixture.
Howe said: 'Callum won't make the game. We're looking and hoping to have him back in full training for the international break, to be ready hopefully for the games afterwards.'