Ameobi is a free agent after leaving Newcastle earlier their summer ending his 14-year association with the Magpies.
A number of English clubs and Major League Soccer sides are thought to be keepings tabs on Ameobi, but Sydney are hoping to win the race for the forward's signature.
The A-League side are believed to have made an offer to Ameobi as they look for a replacement for Alessandro Del Piero.
Ameobi is expected to make a decision on his future and Sydney's offer after playing for Nigeria at the World Cup.
The 32-year-old, who came through the ranks at Newcastle, scored 79 goals in 402 appearances for the club and is sure to have no shortage of admirers this summer.