The Bantams have chalked up four victories from his seven games in charge and the fans have been clamouring for the former England Under-21 boss to be handed a long-term deal.
Taylor has drafted in six new faces at Valley Parade and Lawn is happy with the progress being made under his leadership.
Lawn told the Telegraph & Argus: "I think we will be talking to Peter very soon and seeing what he wants.
"But I need to make it clear that a contract is more than just what wages we are going to pay Peter.
"He's going to want a budget and to make sure we can fulfil his requirements. We've got to wait to sit down and talk to him about that and see where we go from there.
"Talks have started about player budgets, not about his wages.
"He's coming back to us with what he thinks would be a realistic figure to help us get promotion.
"Peter's got it in his own mind who he'd like to keep and who he would let go."
Source: Team Talk
Source: Team Talk