I'm unsure about Tenerife, but I have been to Fuerteventura a few times (7

) and have been deep sea Tuna/Barracuda fishing leaving from Corralejo marina. It's not the cheapest of ventures, but I got a few hours fishing for around 30eu around 4 years ago. There are certainly plenty charter boats leave from here.
There are some wee fish in near the sides which seem to love floating bread and sandy areas are abundant with shoals of fish up to about 1lb, I believe these would come up for a lure or even bread on a hook; they'd be interesting sport on a light fly rod.
The best Tuna area is between Lanzarote and Africa at this time of year, which is only an hour or so from Fuerteventura. It's well worth a go, those fish can really put up a fight

Look out for the Dorado (sp) too, again, they're abundant in the Canaries.
Hope this helps, Gary