Several British media outlets, including the BBC and Sky Sports, reported that the Eagles have had a £35 million bid for the 24-year-old accepted, but that Johnson is yet to green-light the move.
The former Nottingham Forest winger is also attracting interest from Bournemouth, who look set to lose their top scorer, Antoine Semenyo, to Manchester City in the early days of the January transfer window.
"Brennan has been a top professional," said Frank at his pre-match press conference on Tuesday.
"He trained very well. I fully expect him to be available and able to play against Brentford."
Johnson, who cost £47.5 million when he joined from Forest in the summer of 2023, was Spurs' leading scorer last season with 18 goals, including the winner in the Europa League final against Manchester United to end the club's 17-year wait for silverware.
However, he has largely been a back-up to Mohammed Kudus on the right wing since Frank replaced Ange Postecoglou in the close-season.
Johnson has scored four times in 22 appearances in all competitions this season.
Palace boss Oliver Glasner has been vocal about his desire for more forward options.
After a fine start to the campaign, Glasner's side have failed to win in their last five games as a gruelling schedule across three competitions has taken its toll.
"I think, to be honest, if you watch our game, it is pretty clear what we need and then it is a decision of Crystal Palace," said Glasner.
"If you are constantly under-performing in scoring goals, yeah. Look, all I can do is encourage the players, support them and be positive."