Gardner's 85th-minute volley extended the lead Stephane Sessegnon's fine 14th-minute finish had given the Black Cats and killed off any hope of a comeback by the Swans. The midfielder has been linked with a move back to the midlands this month after struggling to settle on Wearside following his £5million summer move.
However, O'Neill is ever more keen for him to stay, saying: "We want him to stay at the football club. At this moment, there has been no concrete offer."
He added: "There is interest from other Premier League teams in and around that area, but there has been nothing concrete and we wouldn't want him to be leaving.
"I have not thought about an immediate replacement at this moment, so time is running out. We want him to stay.
"Obviously at home, he is very family-orientated and he has never really been away before. But it is England and you can get from one place to another. It isn't impossible these days. It's not as if to say he is playing in Saudi Arabia."
Swansea boss Brendan Rodgers had no complaints about his team's performance.
He said: "I thought it was a very, very harsh scoreline. Obviously, in terms of the result, it was disappointing, but the performance level was so high.
"I thought we were fantastic. We created chances and maybe at times our final ball let us down a wee bit, but I can have no complaints. It was hard to take when you play ever so well, but you take your medicine and move on."
Meanwhile, Sunderland striker Nicklas Bendtner will undergo a scan on an eye injury suffered in a challenge with Angel Rangel on Sunday, with O'Neill hoping there is no lasting damage.