The main reason for the idea of their relationship is their on/off screen bond and chemistry with each other and the comedic interaction between them. The fans like to think of a world where they might have a relationship, despite the canon relationships the show has chosen to pursue.
In "The Separation Oscillation", Sheldon suggests that Penny kiss someone else since Leonard was unfaithful to Penny kissing the girl on the boat. Penny agrees and gets into a very passionate embrace with the newly single Sheldon. The whole scenario was a nightmare that Leonard had forcing him to fix the argument he had with his new wife.
In the context of the series, they have been said to have a relationship who's closeness even bothers Amy per "The Viewing Party Combustion". When Leonard and Penny are fighting Leonard mentions the weird relationship she has with Sheldon which she denies. But the Shenny relationship for fans of a romantic bond is purely hypothetical and doesn't necessarily rely on canon, even though they are portrayed almost like brother and sister.
In "The Intimacy Acceleration", Penny and Sheldon conduct an experiment designed to make two people fall in love. The test is loosely based on a theory by psychologist Arthor Aron, called ‘The Experimental Generation of Interpersonal Closeness’ and although they decide the test was unsucessful, they feel more affection to each other and feel closer.
Amy herself proposed the alternate reality where an unattached Penny would go after Sheldon. In "The Cooper Extraction", Shenny has a fantasy sequence about an alternative reality where Penny attempts to seduce Sheldon in the Laundry room. However, this only exists in Amy's mind as she considers the "what ifs" possible if she and Sheldon had never met Leonard.
There is a website for Sheldon and Penny fans called www.thebigbangtheoryhq.com.
Kaley Cuoco has publicly stated that if the two of them got together romantically, Penny would end up killing him. Although most of us believe that Sheldon could very easily hold his own with her and as the show will never test that theory, we may never know.