There are a number of places where you can rent dirt bikes in the southern sinai region of Egypt. If you paid a large enough deposit, they may let you ride it to Nuweiba, jump on the ferry across to Aqaba and ride from there. This shouldn't be too hard, as no carnet is needed for Jordan and you can arrange a temporary import permit at the border. Getting back into Egypt could be a little tricky as customs is a bitch, but with a Egyptian registered vehicle it may be smoother.
You didn't mention if you want to ride off-road or not. Just be aware that the Jordanian authorities try and restrict off-road riding in the protected area of Wadi Rum. It is possible to do it, but you will need to dodge anyone official looking. The area around here is spectacular for experienced desert riders.
For rentals the only place I know that would be contactable on the web is probably Sahara Express, a KTM dealer / renter in Sharm El Sheihk. Try a google search. Otherwise most of the other rental places are small backyard operators in Dahab or Sharm, but some have reasonable 600cc thumpers and old 250 or 400XRs.