McLeish reckons the midlands club will reap the benefits of the Republic of Ireland captain wanting to remain sharp before the Major Soccer League resumes in March and ahead of this summer's Euro 2012 finals.
Villa owner Randy Lerner has sanctioned the deal outside the general sell-to-buy policy to which McLeish has to adhere because of the club's sizeable wage bill.
McLeish, who came close to signing Keane from Tottenham 12 months ago when Birmingham boss, said: "It (the deal) is not far away but I don't like to commit myself until it actually is done.
"The club have sanctioned the deal. They didn't think it was too bad an idea or too big a risk.
"I missed out on Robbie last year although it was probably fortunate because he got injured just after that.
"But Robbie has got good experience, he is good in the dressing room and he is highly motivated.
"He has got a certain competition to play at the end of the season with his country (Euro 2012) so he wants to maintain his level.
"He wants to come here and we would be trying to get things cleared for next week."
McLeish does not expect to be busy this month and has ruled out a move for Celtic midfielder Ki Sung-Yong.
He said: "Am I interested in Sung-Yong? I think that one would be not happening because we can't add to the wage bill."
The Scot has also been linked with a loan move for Chelsea midfielder Josh McEachran.
He said: "It's something we thought about - but I'm not sure we're going to do it."
McLeish has no plans to sell any players this month and downplayed reports QPR have made an offer for central defender James Collins.
He said: "I've no plans to sell anybody. Sometimes, like in my days in Scotland, there was a sacrifice made.
"But at the moment, I've no real plans to let anybody go.
"Have QPR made an offer for Collins? Neil (Warnock) has not called me. Maybe the chief executives have spoken, but I've certainly not had any contact with Neil.
"Normally there are managers' courtesy calls and I've heard nothing."
McLeish once again dismissed reports striker Darren Bent was leaving Villa to join Liverpool.
He said: "We kid Darren along. There's banter like 'when are you off?' and that kind of stuff and he says 'you probably know more than me' and I say 'but, I don't'.
"Again, there's been no contact and he's another one I don't think will be going anywhere.
"If there's any interest out there, I'm not aware of any. We've heard stories saying this club and that club is going to try to buy him.
"But if we say he's not for sale then a) he's not for sale and b) there's been nobody in contact with us."