I have many 2 month old green (unlignified) shoots on an established tree that are available for grafting. I have a dormant lignified cutting that has not rooted after 2 months but still shows green when scraping away some bark. Shoot and cutting are same diameter. Will a graft work? What type is best for this combination?

I have the same situation on some figs & some extra scions.

I was thinking of grafting them (chip or cleft) below the root stock new green growth on last years wood.  May have to move some soil out of the way to get to the old wood.

To your question;
I haven't tried but have heard others have tried green/green grafting.  It seems more delicate an operation but if it takes it should work well.
The caveat is that scion will also need to be new green growth not last year's bud.

I tried dormant cuttings onto growing trees a yr ago. I got one take out of 21 varieties. That one was also the only one that rooted, bad cuttings. I used a splice graft which is like W&T without the tongue. If you can cut it correctly a cleft might be your best bet. Chip budding is another option. That gives one chance for each bud. I find chip harder to fit for good cambium contact vs cleft or splice.

Thanks for the inputs. I decided to tip prune the green shoot I want to graft on to see if that will help it harden up a bit. A very watery looking sap flowed from the cut stem, like 1 part regular milky latex and 3 parts water. I'll post again in a couple of weeks if that helps lignify the shoot.

Try whip and tongue graft,put a plastic bag over the graft to keep the scion moist,place in the shade to avoid too much heat or the graft will fail.This has worked for me with green scion grafted to green rootstock.

"I have a dormant lignified cutting that has not rooted after 2 months but still shows green when scraping away some bark. Shoot and cutting are same diameter. Will a graft work? What type is best for this combination? "
Yes, you can. I would use chip-budding near the base of the green growth where it's hardened a bit more. Don't forget to make a few very shallow cuts below the graft and pinch the new growth.
This will help to remove the excess sap that will drown the graft and make it fail otherwise.

I don't recommend that you even try whip and tongue with a dormant (hardened) scion on a green shoot. It will be almost impossible to execute correctly and the chances of failure are extremely high.

Dormant scion chip bud to a very young green branch is not the best combination (a green chip-bud on a slightly hardened branch has a much better chance) but it can work, if it's the only alternative you have.
As i said, choose the older branch you have or graft the lower you can, searching for the older part of the branch.