Chisora laboured to a victory over Avila but eventually won the match when the Argentinian was unable to continue with 10 seconds left of the penultimate round.
The London heavyweight immediately turned his attention to Liverpudlian Price, with Chisora having designs on an all-English bout this summer.
"I expect him to win the rematch [against Tony Thompson] and me against him would be a great fight for the summer," Chisora told reporters.
"David has nothing to be ashamed of and shouldn't be putting himself down. I like David, I don't have any quarrels with him. He's a family man and a cool guy, but this is a business."
The clash with Avila was the first time Chisora has been in the ring since he was comfortably beaten by David Haye last summer.