The former Liverpool midfielder captained The Magpies under Benitez, who left St James' Park last month after failing to agree on a new contract with the club.
Speaking to the Daily Star on Tuesday, Shelvey said: "I won't be leaving this football club.
"I don't know who wants me to leave or doesn't want me to leave, but as far as I'm aware I'm a Newcastle United player and I've not looked to go anywhere.
"The past season I've never downed tools. I'm here. I love living in the North-East."
Newcastle face a potentially challenging 2019-20 season, with the club yet to make any signings as they begin their pre-season tour today against Wolverhampton Wanderers.