The 19-year-old joined the Clarets from AFC Wimbledon ahead of the 2015-16 campaign and was tipped to make a big impact at Turf Moor.
The youngster has however yet to make a first-team appearance for the Lancashire outfit and he spent the first five months of this season on loan at League One strugglers Coventry, where he scored five goals in 19 appearances in all competitions.
Agyei returned to Turf Moor earlier this month, and it is believed Dyche wants to keep the youngster in his first-team fold, despite reported interest from Ipswich.
One player who could be allowed to leave Burnley however, is on-loan full-back Jon Flanagan, who has struggled to command a regular starting berth since joining the Clarets on a season-long loan from Liverpool during the summer.
The 24-year-old has started just three league matches this season and Dyche has previously admitted that he will monitor the defender's situation over the coming weeks.