The Reds are after a proven centre-back and have been linked heavily with Birmingham's Scott Dann this summer.
However, no deal has materialised for him and now Dalglish is looking elsewhere as he looks for more competition for places in defence.
Dalglish has Jamie Carragher, Daniel Agger and Sotirios Kyrgiakos as options while Martin Skrtel is currently out injured, but he is keen to recruit another centre-back and believes Shawcross could be the answer.
Liverpool have watched the Potters skipper for the past 12 months but have so far failed to come up with an offer suitable to land the 23-year-old, although cash plus Danish international Poulsen may tempt Stoke chief Tony Pulis into doing a deal.
Pulis does not want to lose Shawcross but may be forced into agreeing to the transfer if the player makes it clear he wants to move.