The 29-year-old midfielder agreed a pre-contract agreement to join the Perth side in January, when Steve Lomas was the St Johnstone manager.
With Lomas' departure to join Championship side Millwall, Brittain has had a change of heart, and now wishes to remain in the Highlands, citing 'personal reasons' for his decision.
However, St Johnstone are remaining firm that they wish to keep their man, after a meeting between the two clubs ended in deadlock.
"I've spoken to Roy [McGregor, Ross County chairman] and Richie but nothing's changed. We have no intention of backing down," Saints chairman Steve Brown said.
"We have a new manager in Tommy Wright but that doesn't change Richie's situation. Tommy wants him here for next season and we have Europa League qualifiers in July.
"We offered Richie a two-year deal in January which he grabbed and we expect him to join the squad in pre-season on June 27."