The 19 year old has been linked with a number of Premier League clubs over the last six months, but it appears Swansea have won the race for his signature.
The young playmaker was omitted from City's squad for their defeat to AFC Wimbledon on Sunday and was reported to be training with the Swans ahead of the £1.75m move.
Grimes has previously been likened to former England great Glen Hoddle by Exeter's director of football Steve Perryman. Last season he was voted the club's player of the season and has featured on for the England under-20s on multiple occasions.
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