Manziel was touted to be picked by Jason Garrett's side due to his performances in the state for the Texas A&M Aggies, but despite his links to the club, Jones decided to take offensive lineman Zack Martin.
Jones told NFL Network: "I want you to know that almost as I was handing in the card, it was that close to putting that Manziel card in. It was that close.
"I looked over to my son, Stephen... and I said, 'I took the right pick. [Martin] was the top player by three players on our board at the time that we were sitting there looking at [Manziel]'.
"The question was, 'Could we make it work with Tony Romo during the period of time that we are going to have Tony?' And by the way, my expectations are that Tony will be our quarterback several years to come as we move along."
Manziel was selected by the Cleveland Browns with the 22nd pick in the Draft.