Jose Mourinho has all but confirmed Chelsea will make a third bid for Rooney, but it will not come until after Monday night's game against the England forward's current employers, Manchester United.
Harris, who played 795 times for Chelsea between 1961 and 1980, believes the 27-year-old would be a good signing for the Blues.
"Wayne Rooney is a top-class player. I think the supporters would love him to come," Harris told Sky Sports News Radio.
"I think Wayne's best position is as an out-and-out striker up front on his own.
If he did come, and that's a big if, I'm sure he'd fit in nicely into the system Chelsea want to play.
"He didn't have a particularly good season last year for Manchester United, so hopefully they'll think 'he's only got 18 months left on his contract, it's time to sell him'.
"I don't think his body language looked too clever when the other players were celebrating (at Swansea City) so maybe that tells you his days might be numbered at Manchester United."
Harris, meanwhile, has played down the incident involving Ivanovic and Christian Benteke on Wednesday night.
Aston Villa boss Paul Lambert said it was "clearly an elbow" on Benteke from Ivanovic, who later scored Chelsea's winner, but Harris has pointed to the closeness of referee Kevin Friend.
"Sometimes you get away with things and sometimes you don't, but the referee was near the incident and if he thought it was a deliberate elbow then he would have sent Ivanovic off.
"A lot of people were having a cry about John Terry's hand ball as well but I always believe by the end of the season they level themselves out.
"The same thing might happen next week for Aston Villa, and Paul Lambert will say something completely different."
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