"I would have thought there'd be another two or three coming in soon, there's certainly one in the pipeline," Lawn told BBC Radio Leeds.
"I can tell you lots of players are now ringing Peter Taylor [manager] because they want to play for him."
Central defender Oliver and left-back Threlfall have signed two-year deals at City after successful loan spells at Valley Parade this season, while goalkeeper Saxton has penned a 12-month contract.
"Peter is a wily old fox and wouldn't be drawn on how many players he wants to bring in in total," explained Lawn.
"He knows what he wants and the way he wants to play and knows what players he needs to achieve that.
"Peter seems to be able to improve players and there's been good examples of that at Bradford already and that's why players want to come here."
Lawn is also hopeful that out-of-contract players Michael Flynn, Lee Bullock and Simon Ramsden will soon sign new deals.
"Ramsden and Lee Bullock have strongly indicated that they want to stay here and will sign new deals and I'm confident on them," added Lawn.
"I think Michael Flynn will probably come back too. He's currently on holiday making his mind up. I'm hoping he will come back because I think he's an asset.
"Anyone that knows football and watches Bradford knows that even when Michael has a bad game he is still involved in the game and having a positive impact for us, which is good.
"Let's wait and see what Michael does but I hope he stays here."
Source: BBC Sport
Source: BBC Sport