The 31-year-old netted in a 3-2 victory over his former club West Ham United on Sunday in a rare Premier League appearance for the Merseyside outfit this term.
Rodgers has been linked with deals for Blackpool forward Thomas Ince and Chelsea striker Daniel Sturridge next month, prompting talk of Cole moving to Queens Park Rangers or re-joining the Hammers.
However, Cole is desperate to turn around his career at Liverpool, and the England international hopes Rodgers will give him a sustained run in the first-team.
“All I’m doing is giving my all in training every day, and trying to keep my place in the squad when we play Villa. That’s all I can do,” said Cole.
“It has been difficult recently but I have scored in my last two appearances and I want to keep the sequence going if I can. And when I’m called upon, I will do the best job I can.”
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