From what I can tell both are probably right, although I have doubts about one of my key sources: "The Tuley Family Memoirs", pub. 1906.It states that "Tuley" is derived from "Tulé" from France and that the family fled to the UK (England, Scotland, Ireland) in the 1500's. [I haven't been able to independently confirm this.]
A significant number of Tooley's in American came from some Tuley's [reported to be 3 brothers] who arrived in the late 1600's.The spelling change to Tooley in some lines in the later 1700's and early 1800's with both Tuley & Tooley being used for a while in some lines.
Another group of Tooley's came from Ireland in the 1840's[also reported to be 3 brothers].I suspect that these were descended from the ones who went from France to the UK in the 1500's.
So I guess the answer depends on which line you are from. Even those of us whose Tuley/Tooley line did not go through Ireland can probably easily find ancestors who married people with Irish names.