Were it me,, I would borrow some filter material and about a half cup of aquarium gravel from your 25 gallon tank to the ten gal.(25 gal is cycled Right?)
You could place the borrowed material in a mesh bag, or section of ladies nylon and leave it in the ten gal for around three weeks.
May not be very attractive temporarily,,, but will speed the cycling process.
Once you have a cycled tank such as your 25 gal, there should be no need for cycling another tank when borrowing filter material and gravel or sand from a cycled tank is available.
Will need to keep the borrowed material(bacteria) fed with either pinch of food each day, or by placing two or three small tetra like fishes to the tank and then adding two or three more at ten day intervals.
Keep an eye on water parameter's after adding fish and perform water changes as necessary.