The Sky Blues had warned the academy would be under threat if the rugby club got council permission for the facility at the Alan Higgs Centre.
The football club said it was "not against" Wasps having a facility in the city, but was "bitterly disappointed".
Wasps has said the football club could share the proposed centre.
Coventry City, which leases the land from Coventry Sports Foundation (CSF), has said any reduction in hours its youth players could use facilities may diminish the status of its development programme in the eyes of the Football Association.
Football club chairman Tim Fisher said: "Nobody can tell us what will happen with regard to the scheduling or the facilities that accommodate the academy today.
"We've got nine players in and around the first-team group who all came through the academy."
The planning application was approved with conditions, including over parking and roofing materials.
Wasps Group CEO David Armstrong said it understood "sensitivities regarding the location".
He said the football academy had been offered "use of the indoor kicking barn facility on a commercial basis during set periods" and was keen to hold discussions with the club "about how this may work".
Mr Armstrong said it was open to further talks "with CSF and CCFC about any other ways we might be able to assist" with the academy's future.
A council spokesperson said: "In this case the application seeks to use existing sport and leisure facilities for sport and leisure purposes - planning committee can't make its decision on whether this should be rugby or football.
"If the owners of the site that is currently used by the academy wanted to change that use it would be a private matter between the two parties and is not a planning consideration."
CSF has said it repeatedly invited the football club to meet about the academy's future and that the football club chose not to attend a meeting set up with Wasps last month.
"The offer to sit down and discuss the matter is still very much on the table," a spokesman said.

Wenger, 67, has been criticised by some fans after Arsenal slipped to sixth in the Premier League following four defeats in their past five games.
A 10-2 aggregate loss to Bayern Munich in the Champions League added to the pressure and Wenger says he will make a decision on his future "very soon".
"He's a great man in my opinion," said Iwobi, 20. "I'd like him to stay."
Wenger's contract expires at the end of the season but he has been offered a new two-year deal.
"He's always helped me and developed me as player," Iwobi told the BBC's World Football programme.
"I don't think it's fair the way everybody criticises him. It's very unfair considering what he's done.
"He's always achieved fourth place, been in the Champions League and been up there with the title contenders."
Contract negotiations with two of Arsenal's star names, Alexis Sanchez and Mesut Ozil, have been put on hold until the summer, and Iwobi hopes an agreement can be reached over their futures.
"As you can see from stats, goals and chances they create they are key players in our team," he said.
"Everything offensively almost goes through them. Even though we have other players they are world-class players.
"For the club it'd be great for them to stay, but at the end of the day it's their decisions."
Plans for rugby club Wasps to build a training centre on the site of Coventry City FC's youth academy have been approved.
[NEXT_CONCEPT]
Arsenal forward Alex Iwobi wants Arsene Wenger to remain as manager and says criticism of the Frenchman is "unfair".