The 20-year-old is expected to leave Chelsea this summer with his contract set to expire next season and chances of first-team action minimal.
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Morgan, who can play as a number nine or a number ten, scored 14 goals and provided three assists in his 39 games on loan at Peterborough last season.
According to BBC Sport, West Brom are now progressing in talks with Chelsea as they hope to secure a permanent deal for Morgan.
Chelsea signed Morgan from Southampton as a 17-year-old back in 2022 for a reported fee of £3 million.