The Merseyside outfit are believed to be keen on launching a bid for the England international, who has yet to sign a new deal at the Emirates, in the January transfer window.
"There is nothing to say [about Walcott]," said Rodgers. "It is all speculation about many players. January is always a difficult window. You don't get too many top, top players."
Rodgers, who leads his side into a clash against Newcastle United this Sunday, has been open about his desire to clinch reinforcements in the winter window due to his current depleted squad.