I'm not sure if the OP will come back around, but I went down this rabbit hole again for some unknown reason, so I'll share some of what I found
Thuya burl comes from that area but not 100% sure if it's found in Jordan. Beautiful wood if it does.
I think that most of the thuya burl (sandarac tree) on the market comes from Morocco. It also grows in Algeria and Tunisia, is the national tree of Malta, and also is native to Cartagena, Spain. It is grown as an ornamental, but I couldn't find any reference to Jordan. One thing that is tricky is the various names that are used: sandarac, ghar-ghar, ghar-ghur, gum tree, thyine, and more. I did run across this link for biblical trees that I found interesting:
Daily Bible Study - The Trees of The River . This then led me to another possibility - tamarisk.
It looks like Tamarisk is a common name for a lot of different shrubs. One of these species, Tamarix ramosissima, is known as saltcedar and is an invasive species in the south and western US. Incidentally, I once had a saltcedar blank that I tried to make a pen from, it was looking great until I blew the wood off the tube and it was a complete loss (the wood was solid, my technique not so much). Anyway, it is almost certain that Jordan is part of saltcedar's natural range. I got my blank from Darrin Johnson (Provo, UT) who sells pen blanks on the various facebook marketplaces for woodturners. I'm sure it can be found elsewhere, but I haven't seen it on any of the major wood vendors sites.
"The highlands of Jordan host forests of oak and pine, as well as pistachio and cinnabar trees. Olive, eucalyptus and cedar trees thrive throughout the highlands and the Jordan Valley. Jordan's dry climate is especially conducive to shrub trees, which require less water. Species of shrubs can be found throughout all the geographical regions of Jordan." (Google search).
Acacia also.
Pistachio is a good one that I missed, not widely available but I know that Cookwoods.com has pen blanks available.
Shittum (shittah, shittim, or whatever) is also probably an acacia. The term is used to describe many different species (actually two genus). It is hard to tell what exact species is being sold as acacia or shittum, and then you would have to determine whether or not it is native to Jordan.
Eucalyptus has similar issues to acacia, only a lot more species (only one genus). I don't think that any of the species are native to Jordan or the surrounding area. My understanding is that eucalyptus are only native to Australia and surrounding area -- robuacion could school us on this (if he hasn't already, and he probably has...). Anyway, it is definitely cultivated in Jordan, so might be a good fit.