Reina has failed to impress at Anfield since Rodgers took over as manager at the club.
The Sunday Mirror reports that Rodgers has given Reina until January to show he can fit into the new Liverpool.
Reina has been a key player for the Reds since joining the club from Villarreal for £6million in 2005.
But the Spaniard has made a disappointing start to the season.
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Since Guus Hiddink arrived at the helm, Chelsea have improved considerably after a tricky start to the campaign.
Newcastle United are staring down the barrel of relegation to the Championship for the second time in the past eight years.