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Newcastle manager Glenn Roeder told nufc.co.uk: 'Shay has had surgery to repair a small tear in his lower stomach. He is comfortable and will have a short stay in hospital.
'We'll let things settle down now and it's too early to be putting any date on when he'll be expected back.'
Ireland face Cyprus in Nicosia on October 7 and are then at home to the Czech Republic four days later.
With vice-captain Given seemingly ruled out for six weeks at this early stage, it appears unlikely he will figure in the Group Four matches.
But Staunton, who will next week name his squad, said: 'Shay is still in hospital and we don't yet have a definite timescale for his recovery.
'He is normally a quick healer and all we can do for the moment is monitor the situation closely over the coming days.
'But it would be a real disappointment if he is ruled out of next month's games.'
Recognising what Given offers, Staunton added: 'He's a world-class goalkeeper and has great presence about the squad. He's a leader by nature and any squad would miss his qualities.'