Bramble comes off contract next week.
"It’s never even crossed my mind,” former Sunderland boss McCarthy said. “Titus Bramble will have other clubs interested from him, in the Premier League, and I have not spoken to the player or his agent.
“But I can’t say no because that would be a complete lie.
“What if he came here and said ‘Listen, I love living down here and this is where I come from, what’s your budget, what can you offer me?’, and I said ‘£500 a week’ and he said ‘I love it here and I’ll come and play’?
“I might consider it then under those circumstances. So, if I said no, I’d be lying.”