The Birmingham Mail suggests McLeish has received encouragement from Steve Bruce after enquiring about Jones and that the Trinidad & Tobago international is also keen for the transfer to go through.
McLeish said: "We've made enquiries about a lot of people. There is a lot of transfer speculation now and it is not something I like to talk about when nothing has been done. We have made enquiries for several players. I am not going to confirm or deny any names."
The news comes amid reports that moves for Liverpool's Ryan Babel, Schalke's Kevin Kuranyi and Celtic's Aiden McGeady have all foundered.
Birmingham, who are understood to have a £40 million transfer kitty following Carson Yeung's takeover, made a bid for Babel, believed to be in the region of £8 million, but the clubs are yet to come to an agreement and McLeish could now consider other targets.
A move for Kuranyi is said to have been abandoned after the 27-year-old former Germany striker demanded £70,000-a-week wages, while Celtic are believed to have no interest in allowing McGeady to leave the club.
Various newspapers have suggested that Sunderland could themselves make a £3.5 million move for Kuranyi, which would leave Jones surplus to requirements.