Veliz joins Tottenham's growing injury list ahead of Burnley FA Cup tie
The Argentine was helped off the pitch during the Premier League home win over Bournemouth on New Year's Eve.
"Unfortunately, it was a fairly significant injury," said Postecoglou. "He had scans after the game and it's going to keep him out probably a couple months.
"Won't need surgery but some significant ligament damage, so disappointing for him because he's kind of worked hard to get an opportunity in the last couple of weeks."
Tottenham's South Korea captain Heung-Min Son (30) and Senegal midfielder Pape Matar Sarr (21) will be among the key players missing from Friday's FA Cup third round tie at home to Burnley due to international duty at the Asian Cup and Africa Cup of Nations respectively.
Postecoglou did have some positive news with Dutch defender Micky van de Ven (22) close to returning from the hamstring injury he suffered in their 4-1 Premier League loss to Chelsea on November 6th.
"Micky has trained this week," the Australian said. "I'm not sure about Friday's game, he might be involved in the squad. It is good to get him back but obviously he has missed a fair bit so we will have to be careful with him."
While fifth-placed Spurs are in the thick of the crowded Premier League title fight, six points behind leaders Liverpool, the manager insisted the Premier League does not hold more importance than the FA Cup.
"I know what I feel. Every competition I am in, I want to win. I don't rank them or diminish any against others. Whether it's domestic or continental, everything is hard earned," he said. "I have not sensed anything different with this competition.
"The club has history in this, many of the greatest memories Spurs supporters have will have been in the FA Cup."
Tottenham have won the FA Cup eight times, but their most recent triumph was back in the 1991.
"I'm similar to anyone who loves football. The FA Cup is a significant event. The FA Cup reaches every corner of the globe, it is a great competition with great tradition and I am looking forward to having a crack at it," Postecoglou said.
"At the start of every season every club has ambitions and dreams of winning silverware. More than anything else it is an opportunity for us to try and win a competition and for a club of our stature that should be the ambition every year."