Almost all funk, whether shuffle or straight is written in 4/4 time so this is not really about time signatures but about how to represent this feel in 4/4. If you have a groove where you are playing the first and third notes of 16th note triplets this is most commonly written in 4/4 time with an indication at the beginning of the piece that says “swing 16ths” or “shuffle 16ths”.
There are other ways of representing this literally. One way is to write all 16ths as triplets with alternating 8th and 16th notes. The other solution would be to write in 24/16 with alternating 8th and 16th notes. Neither solution is very practical, especially the second.
Since this type of feel is fairly common and musicians know the difference between straight and swing feels the solution in the first paragraph is acceptable, just like it has been acceptable to write “swing 8ths” or “shuffle 8ths” at the beginning of a piece and use regular 8th notes throughout.