Fremantle coach Ross Lyon has been open about the club's hopes to secure a key forward and said White was on the list of un-contracted players the club was assessing.
"He would be one on the radar but I think everyone would be looking at him," Lyon told 6PR.
"I laid on the couch and watched [the Swans] on the weekend and he played as well as I've seen Jesse play.
"I don't know Jesse at all, but he looks to have matured and been put through football school in a good academy where now he's a very competent player."
"He's certainly on a list of key forward and un-contracted [players]."
White has played six of the last seven games, playing forward and pushing onto a wing where his athleticism is hard to counter.
He kicked an equal career-high four goals against Richmond last Sunday in a superb performance that reignited interest in the 63-game forward.
White, who is out of contract at the end of this season, almost joined Adelaide last October as part of the Kurt Tippett trade, but the deal was spiked at the last minute.
In round 11 he delivered a timely reminder of his ability, kicking three goals and taking seven marks against the Crows in the Swans' 77-point win at AAMI Stadium.
It has been suggested Swans forward Sam Reid, who has kicked seven goals in 10 games this season, could be moved to centre half-back to accommodate White in a full-strength line-up.
White recently said he understood crunch time had arrived if he was to remain with the Swans.
"Talking to my manager about it, it's how I play right now," he said.
"There's some good opportunities for me to play senior footy now and that's all I'm focusing on.
"I know that if I play well and keep producing for the side, good things can come from it."
Fremantle was linked to Mitch Clark when he left the Brisbane Lions, as well as Collingwood star Travis Cloke, Geelong defender Harry Taylor and Richmond Coleman medallist Jack Riewoldt before all re-signed with their current clubs.
Lyon has previously ruled out an attempt to bring Lance Franklin over to the Fremantle in 2014.
Nathan Schmook is a reporter for AFL Media. Follow him on Twitter @AFL_Nathan
