Vertonghen suffered the setback in the first half of last Saturday's Premier League at Huddersfield Town.
The centre-back would have started against Barcelona in the Champions League on Wednesday but now, according to Martinez, faces six to seven weeks on the sidelines.
Tottenham believe the former Everton manager has jumped the gun, claiming that only a second scan due to be made on Monday can present an accurate picture.
And Pochettino made it clear he felt national team managers had an obligation to check with clubs first.
"It's so difficult to communicate with all the national teams," he said. "Look at now, before my press conference you knew about the Vertonghen situation because Roberto told the media in Belgium before us.
"But in the end it's our player, our Tottenham player. Our Tottenham employee. We arrived here and we needed to explain. They need to call us. They need to call and ask. If they call us, we are more than open to talk – and give our opinion."
Tottenham's own opinion turned out to be illuminating enough - it turns out Vertonghen played on at Huddersfield for almost half an hour because he didn't realise how badly he had been hurt.
Assistant manager Jesus Perez said: "We asked him and we thought he could play. The first action was a contact and it's unusual to get a knock in your hamstring. That's why he was playing on. He was assessed at half-time and as soon as all the information was to prevent any further damage, he was off.
Pochettino added: "One component that's important - the adrenaline when you're playing football is so high. And you don't realise. You run under stress, competing. And when you stopping running, you feel it and it's difficult to stop.
"It's difficult to assess during the game because he didn't realise how it was and he only felt a big knock from the striker and he thought it was a dead leg."
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