Villa winger Downing, who will cost around £10 million, is being eyed as a partner for Andy Carroll. Last week the 26-year-old refused to sign a new contract with Villa Park, saying he was "not sure" what he would do next.
Liverpool previously bid £7 million for the services of Blackpool captain Adam in January but the offer was rejected as insufficient. Dalglish apparently views Adam as a "priority" for the club this summer.
The news comes after Liverpool completed a £20 million deal for Sunderland midfielder Josh Henderson.