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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Liverpool have unveiled their new home kit for next season, sticking with the hugely popular retro styling of the current shirt.
The reputation of the Premier League means it is the most popular and one of the most watched leagues in the world.