Kinnear told the club's official website: "Leon Osman is one of a number of players I have enquired about and I'm awaiting an answer from Everton.
"I think I've got to bring in a least three more players. I've got irons in the fire and I'm hoping that this week I'll be able to beef up the squad yet further."
The move has angered Derby boss George Burley who was told by Everton that the midfielder was not available until Moyes strengthened his squad.
Meanwhile, Birmingham and Trinidad and Tobago striker Stern John has again been linked with a move back to the City Ground.
It is believed Joe Kinnear has made enquiries following the £4m arrvial of Emile Heskey at St. Andrews in the close season.
Steve Bruce would prefer a permanent move but a season-long loan deal is most likely for the forward that currently sits behind Clinton Morrison, Mikael Forssell and Emile Heskey in the Blues' pecking order.