Hooker Lake, 26, has 26 caps and made his debut in the 2022 Six Nations opener against Ireland.
Welsh Rugby’s 25-cap rule prohibits players who leave the country to play abroad from representing the national team if they have earned fewer than 25 international caps.
However, their definition of an international cap includes appearances made for the Lions, taking Morgan’s total from 24 up to 26.
“The Ospreys is a club that means a lot to me on so many different levels so making the decision to move was something that I spent a lot of time thinking over,” he said.
“My whole career to this point has been with this team and there have been so many special moments on the way.
“Coming through the pathway, making my debut, the first time I captained the side and the friendships I’ve forged with players and staff, will all live long in my memory for the rest of my life.”
The moves come at an uncertain time for Welsh rugby, with the number of professional clubs in the country set to be reduced from four to three by 2027.
There will be one in Cardiff, one in the east of the country and one in the west – which is expected to result in a straight survival fight between Swansea-based Ospreys and Scarlets in Llanelli.
Another option that has now emerged is Ospreys owners, Y11 Sport & Media, taking over WRU-owned Cardiff, which could produce the desired number of three professional sides.
There are currently more than 80 Wales-based players out of contract at the end of the season.
