Ipswich frontman Wickham, 18, is due to undergo a medical within the next 24 hours after the Black Cats tabled a formal bid of more than 8?million.
In addition, they are expected to hold talks with Birmingham midfielder Gardner, 24, as they try to push home their interest in him.
However, Bruce and chairman Niall Quinn could face a race against time to persuade Wickham that his future lies at the Stadium of Light before his other suitors can stake their own claims.
The player has been heavily linked with Liverpool and is understood to favour a switch to Anfield, while Arsenal and Tottenham also reportedly have an interest in him.
Sunderland will hope they have stolen a march on their potential competitors by getting their blow in first.
Bruce has moved for the teenager in an effort to rebuild an attacking unit which has been torn apart by departures and injury while Gardner could plug the gap in the Black Cats' midfield after Jordan Henderson made his big-money switch to Liverpool earlier this month.