Swansea’s club skipper since 2019, Grimes has been a stand-out performer for the Welsh side since their relegation from the Premier League in 2018 thanks to his quality and durability.
Grimes has made 333 appearances for the Swans, scoring 16 goals, since joining from Exeter City a decade ago this month.
Coventry are 12th in the Championship, four points better off than Swansea, having won three successive league games.
They are six points adrift of the top six with 19 games of the season remaining.
The mid-season transfer window closes on Monday, 3 February.