Blues are in talks with Spurs over a potential £7 million deal to bring the Republic of Ireland international to St Andrew's and McLeish has made it clear signing a forward is his number one priority during the current transfer window.
McLeish said: "It is not fair to name names. We are working actively, the board and myself, to try and enhance the striking positions.
"We know we haven't scored enough goals. That is something we have to rectify in the second half of the season. We know that.
"We know our targets and we've got to go for them."
Blues have also been linked with Cardiff's Jay Bothroyd and - on loan - Liverpool's Ryan Babel.
Keane has slipped down the pecking order at White Hart Lane and is desperate to play regular first-team football. He has also recently been linked with a move to relegation-threatened West Ham although nothing materialised.
Blues' lack of goal power has been the main reason why they have been locked in a survival battle during the first half of the season.
Midfielder Craig Gardner is the top scorer with just five goals and McLeish has become frustrated by his side's inability to convert their chances.
McLeish spent £6million in the summer on Serbia international Nikola Zigic but the former Valencia player is still coming to terms with the demands of the Premier League.
Cameron Jerome has failed to reproduce the form which brought him 11 Premier League goals last season while Kevin Phillips is now at the veteran age of 37.
Keane spent the second half of last season on loan at Celtic.