The former England captain left Anfield for LA Galaxy at the end of last season after scoring 186 goals in 710 appearances for the club.
Despite celebrating his 35th birthday at the end of May, Evans, who was in charge of Liverpool between 1994-1998, is convinced the midfielder ‘still has a lot to offer’.
Speaking to Sky Sports News HQ, he said: “I’m talking as a supporter now and I think they should have tried hard to keep Steven.
“People say they offered him half the contract he had. He gave up England to play for Liverpool but Brendan Rodgers didn’t play him in the Real Madrid game, Man U game, all the big games, he sort of left him out.
“People say he is getting older but for me he still has a lot to offer. I think Liverpool has to forget about the potential in many ways and they have to get players who know they are going to do the job.
“We have a great academy with a lot of lads with good potential coming through the academy and we have to look at them sometimes.
“Jordon Ibe and people like that have signed a contract so hopefully with a mixture of some experience and some younger kids we’ll have a better squad and at least a better attacking option.
“Throughout the season Daniel Sturridge has been injured for most of the time… and sometimes Brendan has played without a striker which has been very difficult for the fans to take.”
Despite failing to build on the previous season’s second-place finish, Evans is convinced Rodgers should remain in charge at Anfield.
“Brendan brought a lot of players in but it didn’t quite work,” he said. “In the end I thought it was quite a disappointing season but I know Brendan recognises that as well.
“A lot of young players, you’re looking for potential but when you are heading to the top end you can only take so much potential, you have to get recognised top players and I don’t think we brought in enough of them.
“At the end of the season I think we paid the price for that. We had a really good chance to make the Champions League with six or seven games to go and with a potential FA Cup final as well.
“But I just think of late Brendan had found it difficult and there was a little bit of a turn from the fans. Sometimes we play without a striker and sometimes fans see players playing out of position.
“I think he should stay and he is there to do the job. But he has to start the season really well otherwise he will be in some trouble.”