The club has been in talks with Odejayi, 35, for a fortnight and are hopeful of securing his signature but manager Craig Elliott says he has other options.
He said: "Talks between us have been quite positive.
"He is a really nice lad and still lives in the area. He is looking to set up a fitness business and a move to us could suit him.
"We are under no illusions and there is another club interested in signing him. We are expecting an answer by the weekend. I think he has been interested in Shaw Lane's story and journey."
Odejayi, who has recently left Stockport, is remembered for his winning goal against Chelsea in the FA Cup and then the subsequent miss against Cardiff City in the semi-final.
Elliott says his signing would have two positives.
He said: "It's a no brainer for us in terms of what he can do on the pitch. But also he has a history with Barnsley FC and hopefully that would generate a bit more interest in the club."