The Blues are believed to have had a previous offer of £16m (€20.2m) rejected by the German side but have not yet been deterred in their attempts to wrap up a deal.
Schurrle made 40 appearances for Bayer last season as they finished fifth in the Bundesliga and reached the last 16 of the Champions League.
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The former Mainz attacker also earned a call-up to Germany's Euro 2012 squad and featured twice as Joachim Low's men reached the semi-final.
Chelsea have adopted a clear policy of buying young talent this summer with the acquisitions of Eden Hazard, his younger brother Thorgan, Marko Marin and Oscar - who are all under 24.
And 21-year-old Schurrle clearly fits into this revolution at Stamford Bridge, which has seen ageing stars such as Didier Drogba and Jose Bosingwa leave the club.