McLeish is normally reluctant to talk about potential transfer targets but he admitted it "could be possible to do some business" regarding Boyd who is out of contract in the summer.
It is understood McLeish would favour Boyd signing a pre-contract with Blues before the January transfer window closes.
The 26-year-old played under McLeish during his spell in charge of Scotland.
If Boyd moves to St Andrews, he will be reunited with his former Rangers and Scotland team-mate Barry Ferguson who has been give a new lease of life under McLeish.
McLeish said: "I can't confirm we have made any offer for anyone. It is not fair to talk about players at other clubs.
"But Boyd is out of contract in the summer and it is possible we could do some business there - but I am not committing myself."
Boyd was also linked with a move to Aston Villa but the Carling Cup finalists have distanced themselves from making a move for the former Kilmarnock player.
McLeish is a big admirer of Gardner who is keen to leave Villa to secure regular first team football.
An initial bid - believed to be around £3 million - was rejected by Villa but McLeish is optimistic a deal will be struck sooner rather than later.
McLeish said: "We are waiting for the go ahead from the bosses (at Birmingham) in terms of the final sanction but we are pretty close to a deal.
"Craig is a player who can perform in a number of positions and a player like that is worth his weight in gold in terms of squad strength."
McLeish refused to comment on whether he is still interested in signing Liverpool striker Ryan Babel, who has healed his rift with Reds boss Rafael Benitez.
Rangers boss Walter Smith confirmed Boyd had more than one offer on the table, amid reports he had also held talks with Aston Villa.
"It is an important decision for Kris," Smith said.
"We have made him an offer - other clubs have made him offers.
"We are hopeful of retaining his services but we just have to wait and see.
"It's a big decision for someone who may feel there is an attraction in going somewhere else.
"We hope the attraction is to stay here and play - as a club, we have made that quite clear."
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