Potteries born midfielder Jack Harrison, who now plays his football in the MLS for New York City, and Galatasaray midfielder Badou Ndiaye also reportedly find themselves on his shortlist.
City have already brought in defensive reinforcements in the form of Moritz Bauer and Konstantinos Stafylidis and now turn their attention towards bulking up the midfield and a possible new goal threat.
Fresh legged finisher
Ings is a talented striker and could really improve Stoke's forward fortunes with his finishing ability which has been lacking of late, with Saido Berahino most notably out of form.
The sort of form that Berahino is in only leaves 36-year-old Peter Crouch as the club's other option, who isn't the same player as he once was.
Ings has only donned the Liverpool shirt 15 times since his move from Burnley as he has suffered two serious knee injuries in consecutive seasons.
He netted 11 goals in the 2014/15 season when the Clarets were relegated which sparked his move to Anfield.
Only one Stoke player has beaten that tally in the last nine seasons, which was Mame Biram Diouf in the same campaign.
From the Big Apple to the bet365
Stoke hope to bring back local lad Jack Harrison, once of the club's academy, but they had an offer of £3.25m for the New York based winger rejected, but the club are believed to be preparing a new offer.
They are eager to complete the move in this window as Harrison has acknowledged his dream of playing in the Premier League.
The winger, who was called up England under-21's recently has a tough decision to make, and swapping Times Square for Stoke City Centre doesn't help.
Lambert has been rumoured to show interest in Senagalese midfielder Badou Ndiaye with an initial offer of £14 million, but the Turkish club are said to have set an asking price of £17.5 million.