I transcript a post of mine from GW forum (it has a few modifications):
I am going to summarize the most important things about grafting figs (according to my experience after several dozens of grafts).
The first thing, you can't expect a near 100% success with figs (unlike apples or pears), a 70 % is a good rate. Figs aren't difficult to graft but they are "capricious", for example, sometimes an apparently successful graft never sprouts. I have a two years old graft of "Dauphine" and I don't achieve to sprout it. I suspect that it depends on the variety too to get success, for example, the past summer I did three grafts of "Sucrette" and all of them quickly thrived , on the contrary, I had to do several grafts of "Sultane" to get a successful one.
I always use the chip budding method for grafting figs. In my climate (oceanic with mild winters and summers), the best months for grafting are May and June.
You can graft in summer too but the grafts don't always sprout before winter. I think that June is the best month for climates with long winters.
Although I usually use a chip of the year (green wood) with a well developed bud and, you can use chip buds from past year cuttings stored in the fridge with good results. In any event you must use wood of the year, not older (when you collect cuttings in winter, choose wood of the year with well developed latent buds).
For the rootstock you can choose a little lignified green wood of the year (don't use immature green shoots) or past year wood, but not older or you will lose your time in the most of cases.
The chip-budding is similar to this:
(Click here) Mundani's site (He is a Majorcan experienced grafter)
in contradistinction to Mundani, I eliminate the leaf petiole, I haven´t found any advantage to keep it.
I cover all the chip except the bud with plastic tape. It is very important to wrap firmly to provide a good contact between chip and rootstock cut surfaces. My graft would be like this:
(this picture is not a real graft, it is only an example)
One of the most critical factors for the first days of a graft is the dehydration so, after wrapping, I cover it with aluminum foil, in this manner I prevent sun burnt and I create a mini "greenhouse" to keep high humidity around the graft.
I do a belly with the aluminum over the bud to leave place for air and for an eventual bud sprouting. I keep the aluminum one week or so.
After several weeks, you have to be prudent to remove the plastic tape, you mustn't be in a hurry, I have seen chips with good aspect to die in few days after removing the tape. The graft union wasn't hardened enough to withstand the drying effect of the air. I wait a minimum of 5 or 6 weeks before removing the tape. In any case, after 3 weeks (it depends of temperatures) you can carefully check the graft estate, if you see abundant callus along the graft union, in few time the graft will can unwrapped.
Later you will can remove definitively the tape with more guarantees, but don't remove before 5 weeks. I have gotten success with fig grafts in only 3 weeks, but a lot of fails too with apparently well joined grafts.
If you see the bud has sprouted and grows well before the 5 weeks, you can unwrap the plastic strip and to wrap again to release a little the plastic pressure and to gain place for the union development. On the contrary, if after 5 weeks the bud hasn't sprouted but it seems healthy, you can keep more weeks the plastic label until it sprouts.
Another important thing is to keep at least one bud above the graft until the sprouting of the graft. Figs are prone to withdraw the sap from grafts if the bud doesn't sprout soon. Keeping at least one bud above the graft you force to the sap to go through the graft, once the graft sprouts vigorously you can cut above it.
Here you can see a cross section of one of my grafts ("oeil dormant" = latent bud). You can see the continuity between cambium layers. This is the key, the cambiums must be in contact! Later, along the season, all the wood above the red line will dry on account of sap retreat. This graft was done the past August and has been latent from then, it has sprouted this spring.
This one is a Kadota graft sprouting in March. As the above one, It was done the past August:
