The 25-year-old Leicester City stopper is one of the players on the Parkhead radar as the Hoops boss seeks to strengthen his squad over the summer.
Daniel Majstorovic, Glenn Loovens and Mark Wilson have been allowed to leave, and Cha Du-Ri and Kelvin Wilson may also head out the door.
Bringing in a solid centre-half is one of Lennon's key priorities and according to Claridge, who played alongside the Celtic boss and Elliott at Leicester, Mills would fit the bill.
The former England under-19 international fell out of favour during the second half of the season after a fall-out with Foxes manager Nigel Pearson, and it is believed he is keen to move on.
However, given that Leicester paid just over £5million for Mills last year, he would come with a significant price tag.
"He is a decent player," Claridge said. "He reminds me a bit of Matt Elliott, who was captain of the Leicester side that Neil and I were in.
"He is a natural leader and a commanding centre-half. I am sure he would have no problems at all fitting in at Celtic.
"He is big, he's strong and he reads the game well. He has to because he is not blessed with a lot of pace, so his positioning and understanding have to be good to compensate for that.
"He is dangerous at set-pieces and can score a few goals. In his own area, he is aggressive when attacking the ball. I can understand why Celtic would like the look of him.
"I am not sure what has happened between him and Pearson, but since the turn of the year we haven't really seen him at all.
"These things happen all the time in football and I am sure that he would like the chance to get out and start over somewhere else.
"Leicester paid just over £5m for him a year ago and for that you would expect him to be a good player."
Claridge, meanwhile, has insisted Celtic and Rangers still carry significant pulling power when it comes to attracting players, despite the turmoil currently afflicting the Ibrox side.
Celtic remain in rude health on and off the park, but events at Ibrox this season have made headlines up and down the country.
It remains to be seen just what lies ahead for Rangers – and the subsequent ramifications for Scottish football – but Claridge believes that any players Lennon may have his eye on will not be put off by the current off-field strife at Rangers.
Certainly, Celtic will have money to invest this summer as they strive to build a squad capable of making it into the group stage of the Champions League and of retaining the SPL title.
"I know about all the off-field trouble involving Rangers, but how many clubs have had financial difficulties lately?" Claridge said. "I think they'll come good again and I don't think that it will put players off the Old Firm. I think Celtic and Rangers are pretty unique. If you know your football, then I think you are always going to sit up and take notice when a club like Celtic are linked with you.
"They have a worldwide reputation. All that history and tradition is quite something and there is a real pull there for most players.
"I'll be honest and insist that I think the standard in the SPL is nowhere near that of the Premiership, and maybe even of the Championship, but I think Celtic and Rangers are a bit different."
He added: "If you can say at the end of your career that you played for one of those teams, it still has a bit of a wow factor.
"Like I said, if you know football, you know about Celtic and you know about Rangers. They are massive. They are brands and in terms of their stature, they are as big as any club in world football."
Celtic have the European carrot to dangle in front of prospective signings, although Claridge is of the opinion that is an added extra.
He said: "Celtic have the chance to play in the Champions League this season, but to be honest I think most players would see that as a bonus rather than a reason for going.
"Even if they get through the qualifiers and into the group stage you wouldn't really expect them to go any further than that.
"The aura of Celtic is what will make them attractive to players. It is a name that excites you and evokes images of what football is all about – the passion, the supporters and the sheer intensity that can surround a team."
However, Claridge won't be shifted from his verdict on the standard of football in the SPL.
Lennon has spoken of his irritation at the manner in which those outwith the Scottish game who see fit to criticise and no doubt he'd have a few choice words to say to his former team-mate.
Claridge said: "I have seen average players in the Championship go up to Scotland and make a name for themselves and I think Matt would have no trouble holding his own for Celtic.
"It is fair to say that the SPL is not a great standard and that is what puts people off. But I think you have to bracket Celtic and Rangers a little differently simply because their names still carry a lot of weight."