Fifth-placed Spurs are still clinging on in the battle to finish in the top four in the Premier League. They are on 60 points, seven behind Aston Villa, with two games in hand in the race for Champions League qualification.
Forward Timo Werner has also been ruled out of Thursday's trip to his former club Chelsea, who are ninth, 12 points behind Spurs.
The 28-year-old German will miss the remaining games of the season after being injured against Arsenal on Sunday.
"Timo (Werner) with his hamstring - he's only got two-and-a-half weeks to go so he will miss the rest of the season," the Australian manager told reporters.
"Ben Davies got a calf injury from the (Arsenal) game and he'll miss the rest of the season as well."
Werner has scored two goals and registered three assists in 13 league games since signing on loan in January from Bundesliga side RB Leipzig. Versatile Welshman Davies has made 19 appearances for Spurs this season.
Tottenham are hoping to bounce back from Sunday's 3-2 home defeat by Arsenal but face tough challenges ahead with fixtures against Liverpool and Manchester City to follow after the Chelsea clash.
Postecoglou's side had started the season strong but suffered their first loss at home to Chelsea in November. Asked if his side had shown progression in his first year in charge, the coach said: "Clearly, as clear as you want it to be, 20-20, definitely, 100 per cent.
"What I see is us playing football, measuring ourselves against the best. My players have got real belief in what we are doing. That's all I need to see."
Attacking midfielder James Maddison was injured in the home defeat by Chelsea and since returning in January he has struggled to reach the heights he hit in Tottenham's impressive start to the campaign.
Asked about Maddison's form, Postecoglou said: "We've had a very disruptive season. He's another one that's had disruptions. First 10 games, he played every game and he was going really well.
"Then he picked up an injury, missed a lot of football and he's come back but we've had challenges as a team.
"We haven't had any real fluency in terms of cohesion, games and starting line-ups. He's had to battle through that like a lot of them."
