The Northern Ireland defender is expected to be drafted straight back into the side for Sunday's Premier League trip to Manchester City.
The return to fitness of Evans comes as makeshift centre-back Wilfred Ndidi is set to be out for up to three months with an abductor injury which will need surgery.
Evans, 32, has now served a suspension following his sending-off against Manchester United at the end of last season and recovered from a thigh strain picked up while on international duty.
Rodgers is keen for the former United and West Brom defender to remain part of the club's long-term plans, with his current deal set to expire in the summer of 2021.
"Our intention was to get through to the end of the season then the club and Jonny's agent would pick it up from there," Rodgers said.
"He is clearly a player I would love to keep here. He has been absolutely brilliant, not just in the games but in training with setting the standards and the mentality.
"I believe the discussion will go on between Jonny's agent and the club, and hopefully we can find a solution to keeping him here."
Allowing Evans to depart should an unexpected bid come in before the end of the transfer window is "absolutely not" in the thinking of the Leicester boss.
"I have always said he has got a brain of the very top level, Jonny, and has great experience," Rodgers added.
"He really helps the players around him when he is playing. To have someone of that calibre around the training ground and in the games for us is very important.
"You can never underestimate his qualities and what he brings to us as a team, how we perform, and I am just glad he is back."
Rodgers hopes Ndidi will come return stronger following his latest injury blow.
"To be out having had a period out last season is obviously disappointing for him," the Leicester boss said.
"We know how he recovers and his determination. We think after the operation, with the type of injury it is, we think he should be able to get out onto the pitch in about six to seven weeks, which then allows him to build up his fitness.
"It is a huge blow, of course it is, because he is one of the best defensive midfield players in the league and is so important with how we work, but it is also now an opportunity which will be there for someone else."