The Argentine striker has been linked with a January move to a Premier League club after scoring 19 goals for Leeds this season.
Speculation increased when Becchio missed the 2-1 win over Birmingham in their third-round replay, but Warnock insisted he was left out due to illness.
"I've heard the rumours but I think he has texted Shaun Harvey (Leeds chief executive) to say he is at home," said Warnock. "So he's obviously heard the rumours.
"We lost Becchio this morning, he came off training on Monday and said he felt sick. And then the doc said he was sick in the night so he didn't travel."
Warnock also confirmed he was not aware of any bids for Becchio, but when asked if he wanted to stay at Elland Road, he added: "We'll see. I can't tell you anything other than that.
"I spoke to the owners and in due course we'll probably speak about that."
Leeds trailed at half-time at St Andrew's but hit back after the break as goals from Ross McCormack and El-Hadji Diouf booked a home tie with Tottenham.
The team had come under heavy fire following their weekend drubbing by Barnsley, and Warnock added: "We showed character and I'm really proud of them.
"They're the best type of results when everyone writes you off. They gave everything in the second half. Everyone contributed.
"It's a great tie in the next round - Leeds v Tottenham, it whets the appetite."