Johnson finished as Spurs' leading scorer last season under the club's Australian former boss, Ange Postecoglou, with 18 goals.
He also scored the winner in the Europa League final against fellow EPL side Manchester United to end the club's 17-year wait for silverware.
Johnson was a key early signing from Nottingham Forest by Postecoglou, and his close-range goal clinched victory in the final in Bilbao in May.
That triumph wasn't enough to save Postecoglou from the sack, though, and Johnson has been a peripheral figure this season under his successor, Thomas Frank.
Johnson has made just six Premier League starts under Frank, and has not scored a league goal since the opening two games of the season.
However, Frank insisted Johnson could still play for Spurs at Brentford on New Year's Day.
"I think Brennan has been a top professional, is a top professional, trained well today. He's ready for two days' time."
Johnson has largely been a back-up to summer recruit Mohammed Kudus on the right wing and now Palace are leading the race to sign the 24-year-old after agreeing a £35m (A$70m) deal.
Johnson is still deciding on whether to move and there are other suitors, including another EPL side in Bournemouth, and uncertainly over the long-term future of Palace boss Oliver Glasner.
Glasner has made clear on multiple occasions his frustration over a lack of transfer business last summer when Eberechi Eze was sold to Arsenal midway through August.
"To be honest, if you watch our game, it is pretty clear what we need and then it is a decision of Crystal Palace," he said.
The transfer window is open in the UK from January 1 to February 2 and Bournemouth's highly-rated winger Antoine Semenyo could be among the first big deals if, as expected, the Ghana forward joins Manchester City.
With agencies
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