Webber is available on a free transfer after leaving Bramall Lane and has been attracting interest from several clubs.
Warnock had been keen to sign the 27-year-old after working with the former Manchester United youngster during his time in charge at Bramall Lane.
However, Warnock is resigned to missing out on Webber and believes he will sign for Boro following their relegation from the Premier League.
"It looks as if Danny will be going to Middlesbrough - it's difficult for us to compete with their money," Warnock told the Croydon Advertiser.
He added on Sky Sports News: "Danny is a good player and I know he has got one or two offers, which I'd expect for a player of his quality."
Middlesbrough have held talks Webber and Boro boss Gareth Southgate told the Evening Gazette: "We were made aware of Danny's availability as a free agent and we have held talks with him."
Warnock saw Serbian trialist Goran Maric score in Thursday evening's 4-0 friendly win over Boreham Wood and confirmed he is looking to sign a striker with a physical presence.
"I need a taller, bigger lad to start with," said Warnock. "We will have to see in the next few weeks when we probably come back from America as to what's available and what we're going to go for.
"We need a bit of luck, like anybody else."