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The 32-year-old, who had an outstanding Euro 2016, is a free agent after leaving Hamilton Academical at the end of last season.
Southampton were reported to be keen to sign McGovern but he is expected to finalise a deal with Norwich this week.
The keeper had previously worked under Norwich boss Alex Neil at Hamilton.
John Ruddy and Declan Rudd are the current senior keepers at Carrow Road.
McGovern, from County Fermanagh, played in Northern Ireland's four matches at the European Championships.
He earned rave reviews for a remarkable display when Northern Ireland restricted Germany to a 1-0 victory in Paris, a result good enough to send Michael O'Neill's men into the knockout stages.
As well as Southampton, Burnley were also reported to be interested in signing McGovern.
Neil last week helped his former club replace McGovern, who rejected a new contract, by sending goalkeeper Remi Matthews to Accies on loan for the season.
Championship club Norwich City have emerged as the favourites to sign Northern Ireland goalkeeper Michael McGovern.