The 21 year-old Ireland under 21 international has yet to start a first-team game for the Hammers, but did play seven matches on loan at Lincoln City in the 2019-20 season.
Coventry has impressed for Hammers in their pre-season games this summer and scored a goal in a 2-1 win over Northampton Town.
But Hammers boss David Moyes is not sure he will break into his first-team squad so could be prepared to let him go, possibly on loan
Coventry is described as a ‘deep-lying’ midfielder which is exactly the position Posh are seeking to fill.
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Coventry City and Hull City have also been linked with the player, an international teammate of current Posh midfielder Jack Taylor.
Posh boss Darren Ferguson expects a new defensive midfielder to arrive at the club before the Championship opener at Luton on August 7.