Sydney CEO Tony Pignata confirmed on Thursday that the Cottagers had contacted the club with a view to bringing the 19-year-old midfielder to London for a trial in January in what could be a prelude to a permanent switch during the transfer window.
But with rock bottom Sydney desperate for points and needing to field its strongest possible team in the coming weeks, the club will have reservatiions in letting him go.
"I will sit down with Terry, his father and the chairman (Scott Barlow) and discuss the situation," said Pignata.
"Fulham have been in touch and want to have a look at him. Anything is a long way from being decided right now and the situation is complicated by the fact that Fulham would want him there in January and we also need him for a number of crucial matches coming up.
"It's a bit of a Catch 22 at the moment because he's important for us."
A breakthrough performer for Australia at the recent East Asian Cup qualifiers, Antonis is out of contract at the end of April and has made no secret of his desire to test himself overseas.
Fulham is already home to Australians Mark Schwarzer and development player Ryan Williams, and Pignata admitted it would be difficult to stand in Antonis' way were a realistic offer forthcoming.
"if it turns out to be a good opportunity for the lad, and it’s the right thing for all parties, then we could make it work," he said.
'That said, we are not keen to lose him at all and we are in further discussions over a contract extension.
'Terry is a very talented player and we would prefer that he stayed with us."
Antonis is focusing on now, rather than the future, explaining: 'My main focus is trying to get to the semi-finals with Sydney FC and everything else happens after that.
'I am not looking too far ahead right now."
He is convinced that bottom-placed Sydney is far from out of A-League title calculations and will begin its resurgence against the Newcastle Jets at Hunter Stadium on Saturday night.
'We are really unhappy at losing the derby against Western Sydney Wanderers but it hasn’t knocked our confidence and we will turn things around," he insisted.
Though a whopping 16 points adrift of leader Central Coast, Sydney is only three points out of the top six and Antonis believes the clash against Newcastle could prove a catalyst.
'We obviously owe them one for coming to our place and beating us 3-2 in round two. We are not letting ourselves get down over our position in the table," Antonis continued.
'The boys are keeping their heads up high and we believe things can and will change very quickly.
'Our main focus now is to get to the semi-finals and I believe we will do that. We still have a great roster of players and I believe in our ability to put things right."