Both clubs have been chasing the 19-year-old star but sources close to the Belgian club claim Chelsea have made the first bid.
If the Blues are successful, Courtois is expected to remain in Belgium on loan next season, before making the switch to Stamford Bridge in 2012.
In demand: Thibaut Courtois looks set for a move to the Premier League
Courtois has been an ever present in the Genk side that marched to the Belgian title, and is seen as the eventual successor to Chelsea's current No 1 Petr Cech.
With Shay Given likely to leave City in the summer, Roberto Mancini is on the lookout for backup to Joe Hart.
Benfica and Schalke were also tracking the 6ft 4in shot stopper.
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Explore more:People: Shay Given, Roberto Mancini, Joe Hart Places: Belgium