The Reds chief is expected to invest heavily in his first-team squad at the end of the season, with Champions League football at Anfield looking more and more likely as the weeks pass.
Rodgers is thought to be looking for a new goalkeeper and a left-back, and sees Swans pair Davies and Vorm as ideal candidates.
The Northern Irishman did a sterling job at the Liberty Stadium prior to taking charge of Liverpool, and has already returned to his former club to snap up Joe Allen.
He is now said to be ready to do so once again for the star pair, who may both be tempted by a switch to Merseyside.
Vorm is believed to be keen to step up to the next level after enjoying a good spell in South Wales, and could be lured away from the club with the promise of European football.
Davies has already confessed that he is happy at his boyhood club, but the 20-year-old may find turning down a bigger side difficult as he looks to establish himself in the Premier League.
Liverpool may also make a fresh move for Dnipro Dnipropetrovsk wide-man Yevhen Konoplyanka – whom they almost signed in January.