Palestinian activists have set up a Facebook campaign to "abduct a partner for Shalit".
The page, set up earlier this week, calls for the capture of a female prisoner to use as bait so that the Israeli authorities will release Palestinian prisoners.
The group's profile photograph has been edited to show an Israeli soldier chained and held by an armed woman in a veil.
Members have posted antisemitic taunts, with one writing: "Soon all the people of Israel will become like Shalit. Just wait. Shalit engagement, congratulations to you!"
He also added that the pair should have children together.
Another user, Yasseen Elsharef, agreed: "Would be a wonderful wedding."
The campaign comes almost five years after Corporal Shalit was captured by Hamas in a cross-border raid near the Kerem Shalom crossing.
He has been kept hostage at an unknown location in the Gaza Strip since then, with Hamas demanding that Israel release thousands of prisoners in return for letting him go.