Adams, 24, represented England at under-20 level before opting to play for the Scots earlier this year – qualifying through a maternal grandparent.
He has already scored two goals for the Tartan Army in four internationals.
Asked what it was like to play up front for your country, Saints boss Hasenhuttl told the Daily Echo: “The spot is on you. That is a fact.
“If you are performing well, you're the hero. If not, then it's tough I think. This is when you play for your country – the whole nation is looking on you.
“But I think he can handle this and he has the quality to compete in the Euros. I wish him all the best when he's part of them.
“I am sure that he will score his goals."