The Reds chief is desperate to add to his midfield options despite claims that there will be no major arrivals at the club this month, with a series of injuries having exposed a lack of depth.
Gomes is highly-rated in Lisbon, but has struggled for first-team opportunities this season due to the presence of Nemanja Matic.
But with the Serbian ace now a Chelsea player, Gomes may get more opportunities to impress.
However, Liverpool are ready to make a serious offer to lure him to Anfield.
Rodgers sees Gomes as an ideal addition to his midfield, after growing aware that ageing skipper Steven Gerrard may be unable to play at the highest level on a weekly basis.
Benfica are said to be willing to do business for around £12m, which is a figure the Northern Irishman may be ready to stretch to after seeing a deal for Mohamed Salah collapse.
Liverpool are also watching Manchester United target Saul Niguez.