O'Donovan has scored eight goals in his last 12 games, including two in Saturday's 3-1 defeat of Colchester at Victoria Park, and his contract with the Black Cats expires at the end of the season.
Turner told the Hartlepool Mail: "I've recommended to the club that we sign him. But he's out of contract at the end of the season and the more goals he scores, the more options he might get.
"To be in the top half consistently and be in the top eight, you have to have someone up front who can score you 20 goals a season at least. If Roy was here at the start of the season coming into games now, I think he'd be on 18, 20-odd goals.
"He's done superb for us. It's not just his goals. His work-rate, commitment and attitude is top draw."