Agyei, 19, joined the Clarets from AFC Wimbledon in the summer of 2015 but is yet to feature at first-team level for the Lancashire club.
"He's a player with huge potential and we are confident he can make a positive contribution to our team," Sky Blues boss Tony Mowbray said.
"I'm looking forward to working with him during his spell at the club."
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Coventry City have signed striker Dan Agyei on loan from Premier League side Burnley until 17 January.