The duo will be heading for the exit door at the end of the season along with Sam Hidalgo-Clyne and Cornell du Preez.
When Scotland cap Weir headed along the M8 from Glasgow Warriors to Edinburgh, many saw it as the start of a renaissance for the club and the player as he aimed to regain his international spot.
However, it has never quite happened for Weir in the capital, the 26-year-old seeing the head coach that signed him, Alan Solomons, leave soon after and then suffering a number of injuries.
As a result, he was due to start a loan at Worcester Warriors this week, but has been recalled by Edinburgh due to a lack of backs cover.
Cockerill said: “We have Simon Hickey coming to play stand-off and Jaco van der Walt has come in and done well, so a cost comes with those two and we don’t need three 10s because there is not enough space at the table playing wise or financially.
“The way we want to play suits different personnel.”
Welsman Tovey has been at the club since joining on loan from the Cardiff Blues in 2016. And, with a quartet of departures, Cockerill has moved to calm supporters fears about the quality of the squad for 2018/19.
“We are looking to build the squad over the next three, four or five years to have some continuity and build some strength so we are able to compete,” he said.
“This is not about any one individual and if nothing changes then nothing will change. When I arrived this team had not done well in years so something had to change. Now we have changed things I feel we are moving in the right direction.
“We are playing a good brand of rugby, we are bringing through some good young Scottish lads, we are getting the best out of what we’ve got and I am excited about the future.”
So, with spaces freed up in the squad, could Edinburgh be about to re-sign Scotland centre Matt Scott?
He was a fans’ favourite between 2011 and 2016 and has been told he can leave by Gloucester, but Cockerill is remaining tight lipped. “I’m not in a position to confirm that at this point,” he said.
“But we’re looking to strengthen the squad – we’re looking to strengthen the backline and hopefully there will be some positive news around quality players coming into the team soon.
“The squad will be smaller next year, but I think it will be a better quality.”