I don't really think so...
The largest a gas giant can get without being a star itself is a L class brown dwarf, which is just under the size needed for fusion to start.
The least massive star known is 2MASS J0523-1403, which is just over the fusion threshold.
You can get smaller stars (in radius), like white dwarfs, or neutron stars, but they are going to have a lot more mass.
A brown dwarf just under the limit is going to be larger than a red dwarf just over the limit, but that's because as they get more massive the density increases.
Adam Wood's answer got me thinking: if the gas giant had a massively oversized iron core, that might be able to stop the fusion process from starting up, as iron doesn't fuse easily because the process is endothermic.
I don't know how big of an iron core you'd need to keep something with one solar mass from beginning fusion, or how much mass it could have before it started to do weird things...
Edit 2: Weird things
Apparently, according to this answer, a iron sphere can be up to 1.44 solar masses before the gravitational pressure would cause it to collapse into a neutron star.
This gives you a lot of room for a small start to orbit it.
So, if you had an iron core that was around .5 or so solar masses and then surrounded by thinnish layer of hydrogen gas so that it stayed under the fusion limit, it could be orbited by a red dwarf like 2MASS J0523-1403 (0.08 solar mass range) easily.